Calculus Maximus WS 7.3 Separable Diff EQ basd.net. solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example., Examples Solve the (separable) differential equation Solve the (separable) differential equation Solve the following differential equation: Sketch the family of solution curves. Videos See short videos of worked problems for this section. Quiz. Take a quiz. Exercises See Exercises for 3.3 Separable Differential Equations (PDF). Work online to solve the exercises for this section, or for any.

### Section 9.3 Separable Equations University of Portland

Separable equations introduction Differential equations. Separable Differential Equations We start with the deп¬Ѓnition of a separable diп¬Ђerential equation. Deп¬Ѓnition 1.1. A separable equation is a п¬Ѓrst order diп¬Ђerential equa- tion in which the expression for dy/dx can be factcored as a function of x times a function of y. In other words, it is an equation of the from dy dx = g(x) f(y) (we write it as a fraction for convenience). To solve, An example of a linear equation is because, for , it can be written in the form Notice that this differential equation is not separable because itвЂ™s impossible to factor the.

Basics and Separable Solutions We now turn our attention to differential equations in which the вЂњunknown function to be deter- minedвЂќ вЂ” which we will usually denote by u вЂ¦ Example 1: Solve and find a general solution to the differential equation. y ' = 3 e y x 2 Solution to Example 1: We first rewrite the given equations in differential form and with variables separated, the y's on one side and the x's on the other side as follows.

Suppose we have the п¬Ѓrst order diп¬Ђerential equation P(y) dy dx = Q(x) where Q(x) and P(y) are functions involving x and y only respectively. For example y2 dy dx = 1 x3 or 1 y2 dy dx = xв€’ 3 x3. We can solve these diп¬Ђerential equations using the technique of separatingvariables. General Solution By taking the original diп¬Ђerential equation P(y) dy dx = Q(x) we can solve this by A separable differential equation is a common kind of differential equation that is especially straightforward to solve. Separable equations have the form \(\frac{dy}{dx}=f(x)g(y)\), and are called separable because the variables \(x\) and \(y\) can be brought to opposite sides of the equation.

An example of a linear equation is because, for , it can be written in the form Notice that this differential equation is not separable because itвЂ™s impossible to factor the A п¬Ѓrst-order differential equation is called separable if it can be written in the form p(y) dy dx = q(x). (1.4.1) The solution technique for a separable differential equation is given in Theorem 1.4.2. Theorem 1.4.2 If p(y)and q(x)are continuous, then Equation (1.4.1) has the general solution p(y)dy= q(x)dx+c, (1.4.2) where c is an arbitrary constant. Proof

A first order differential equation \(yвЂ™ = f\left( {x,y} \right)\) is called a separable equation if the function \(f\left( {x,y} \right)\) can be factored into the product of two functions of \(x\) and \(y:\) Exact Differential Equations вЂў Integrating Factors Exact Differential Equations In Section 5.6, you studied applications of differential equations to growth and decay problems. In Section 5.7, you learned more about the basic ideas of differential equa-tions and studied the solution technique known as separation of variables. In this chapter, you will learn more about solving differential

For similar discussion and examples, see David LomenвЂ™s article вЂњSolving Separable Differential Equations: Antidifferentiation and Domain Are Both NeededвЂќ in the Course Home Pages section of AP Calculus at the AP Central website. Example 1: Solve and find a general solution to the differential equation. y ' = 3 e y x 2 Solution to Example 1: We first rewrite the given equations in differential form and with variables separated, the y's on one side and the x's on the other side as follows.

Section 2-3 : Exact Equations. The next type of first order differential equations that weвЂ™ll be looking at is exact differential equations. Before we get into the full details behind solving exact differential equations itвЂ™s probably best to work an example that will help to show us just what an exact differential equation вЂ¦ A separable differential equation is a common kind of differential equation that is especially straightforward to solve. Separable equations have the form \(\frac{dy}{dx}=f(x)g(y)\), and are called separable because the variables \(x\) and \(y\) can be brought to opposite sides of the equation.

View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦ Advanced Math Solutions вЂ“ Ordinary Differential Equations Calculator, Bernoulli ODE Last post, we learned about separable differential equations. In this post, we will learn about Bernoulli differential...

Separable differential equations can be described as first-order first-degree differential equations where the expression for the derivative in terms of the variables is a multiplicatively separable function of the two variables. View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦

Linear Differential Equations web.stanford.edu. Separable Differential Equations We start with the deп¬Ѓnition of a separable diп¬Ђerential equation. Deп¬Ѓnition 1.1. A separable equation is a п¬Ѓrst order diп¬Ђerential equa- tion in which the expression for dy/dx can be factcored as a function of x times a function of y. In other words, it is an equation of the from dy dx = g(x) f(y) (we write it as a fraction for convenience). To solve, S. Ghorai 1 Lecture III Solution of rst order equations 1 Separable equations These are equations of the form y0= f(x)g(y) Assuing gis nonzero, we divide by gand integrate to nd.

### y Separable Differential Equations Calculator - Symbolab

Separable Differential Equations Scribd. 4. DIFFERENTIAL EQUATIONS 4.1: CONSTRUCT THE DIFFERENTIAL EQUATIONS 4.1.1: Identify Type Of Differential Equations Order в†’ The number of the highest derivative in a differential equation., An example of a linear equation is because, for , it can be written in the form Notice that this differential equation is not separable because itвЂ™s impossible to factor the.

### Separable Differential Equations Calcworkshop

Separable Diп¬Ђerential Equations University of British. Separable Diп¬Ђerential Equations A diп¬Ђerential equation is an equation for an unknown function that involves the derivative of the unknown function. https://en.wikipedia.org/wiki/Separable_ordinary_differential_equation Answer interactive questions on separable differential equations. See what you know about specifics like how to solve a differential equations with 0 as a variable and how to identify a separable.

Basics and Separable Solutions We now turn our attention to differential equations in which the вЂњunknown function to be deter- minedвЂќ вЂ” which we will usually denote by u вЂ¦ Separable means that we can keep those two separately and do an integral of f and an integral of g and we're in business. OK. Examples. Suppose that f of y is 1. Then we have this simplest differential equation of all, dy/dt is some function of t. That's what calculus is for. y is the integral of g. Suppose there was no t. Just a 1 over f of y, with g of t equal one. Then I bring the f of y up

Separable equations are the class of differential equations that can be solved using this method. "Separation of variables" allows us to rewrite differential equations so we obtain an equality between two integrals we can evaluate. We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at вЂ¦

S. Ghorai 1 Lecture III Solution of rst order equations 1 Separable equations These are equations of the form y0= f(x)g(y) Assuing gis nonzero, we divide by gand integrate to nd For similar discussion and examples, see David LomenвЂ™s article вЂњSolving Separable Differential Equations: Antidifferentiation and Domain Are Both NeededвЂќ in the Course Home Pages section of AP Calculus at the AP Central website.

Mixing Tank Separable Differential Equations Examples When studying separable differential equations, one classic class of examples is the mixing tank problems. Here we will consider a few variations on this classic. Example 1. A tank has pure water п¬‚owing into it at 10 l/min. The contents of the tank are kept thoroughly mixed, and the contents п¬‚ow out at 10 l/min. Initially, the tank Answer interactive questions on separable differential equations. See what you know about specifics like how to solve a differential equations with 0 as a variable and how to identify a separable

A first order differential equation \(yвЂ™ = f\left( {x,y} \right)\) is called a separable equation if the function \(f\left( {x,y} \right)\) can be factored into the product of two functions of \(x\) and \(y:\) 25/08/2011В В· A basic lesson on how to solve separable differential equations. Such equations have important applications in the modelling of dynamic phenomena. Such equations have important applications in the

For similar discussion and examples, see David LomenвЂ™s article вЂњSolving Separable Differential Equations: Antidifferentiation and Domain Are Both NeededвЂќ in the Course Home Pages section of AP Calculus at the AP Central website. Suppose we have the п¬Ѓrst order diп¬Ђerential equation P(y) dy dx = Q(x) where Q(x) and P(y) are functions involving x and y only respectively. For example y2 dy dx = 1 x3 or 1 y2 dy dx = xв€’ 3 x3. We can solve these diп¬Ђerential equations using the technique of separatingvariables. General Solution By taking the original diп¬Ђerential equation P(y) dy dx = Q(x) we can solve this by

PaulвЂ™s Online Notes, emphasizes this fact when stating that for a differential equation to be separable, all the yвЂ™s in the differential equation must be multiplied by the derivative, and all the xвЂ™s in the differential equation must be on the other side of the equal sign. Suppose we have the п¬Ѓrst order diп¬Ђerential equation P(y) dy dx = Q(x) where Q(x) and P(y) are functions involving x and y only respectively. For example y2 dy dx = 1 x3 or 1 y2 dy dx = xв€’ 3 x3. We can solve these diп¬Ђerential equations using the technique of separatingvariables. General Solution By taking the original diп¬Ђerential equation P(y) dy dx = Q(x) we can solve this by

Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria. 4. DIFFERENTIAL EQUATIONS 4.1: CONSTRUCT THE DIFFERENTIAL EQUATIONS 4.1.1: Identify Type Of Differential Equations Order в†’ The number of the highest derivative in a differential equation.

Determine whether each of the following differential equations is separable or not. a constant of integration is always present. 1) gives 1 = B 1^2/3 = B. We will use the general solutions from the previous examples. y) = (1. Simplifying this gives C = 1. y) = (1. Substituting (x. PaulвЂ™s Online Notes, emphasizes this fact when stating that for a differential equation to be separable, all the yвЂ™s in the differential equation must be multiplied by the derivative, and all the xвЂ™s in the differential equation must be on the other side of the equal sign.

## Separable First Order Differential Equations Basic

Separable differential equation Calculus. Basics and Separable Solutions We now turn our attention to differential equations in which the вЂњunknown function to be deter- minedвЂќ вЂ” which we will usually denote by u вЂ¦, Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria..

### Exact Differential Equations Cengage

Differential Equations Equations Differential Calculus. solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example., Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria..

For similar discussion and examples, see David LomenвЂ™s article вЂњSolving Separable Differential Equations: Antidifferentiation and Domain Are Both NeededвЂќ in the Course Home Pages section of AP Calculus at the AP Central website. For similar discussion and examples, see David LomenвЂ™s article вЂњSolving Separable Differential Equations: Antidifferentiation and Domain Are Both NeededвЂќ in the Course Home Pages section of AP Calculus at the AP Central website.

Separable differential equations can be described as first-order first-degree differential equations where the expression for the derivative in terms of the variables is a multiplicatively separable function of the two variables. Differential equations arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in вЂ¦

Separable equations are the class of differential equations that can be solved using this method. "Separation of variables" allows us to rewrite differential equations so we obtain an equality between two integrals we can evaluate. solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example.

1/02/2017В В· This calculus video tutorial explains how to solve first order differential equations using separation of variables. It explains how to integrate the function to find the general solution and how Answer interactive questions on separable differential equations. See what you know about specifics like how to solve a differential equations with 0 as a variable and how to identify a separable

Examples Solve the (separable) differential equation Solve the (separable) differential equation Solve the following differential equation: Sketch the family of solution curves. Videos See short videos of worked problems for this section. Quiz. Take a quiz. Exercises See Exercises for 3.3 Separable Differential Equations (PDF). Work online to solve the exercises for this section, or for any Mixing Tank Separable Differential Equations Examples When studying separable differential equations, one classic class of examples is the mixing tank problems. Here we will consider a few variations on this classic. Example 1. A tank has pure water п¬‚owing into it at 10 l/min. The contents of the tank are kept thoroughly mixed, and the contents п¬‚ow out at 10 l/min. Initially, the tank

4. DIFFERENTIAL EQUATIONS 4.1: CONSTRUCT THE DIFFERENTIAL EQUATIONS 4.1.1: Identify Type Of Differential Equations Order в†’ The number of the highest derivative in a differential equation. solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example.

4. DIFFERENTIAL EQUATIONS 4.1: CONSTRUCT THE DIFFERENTIAL EQUATIONS 4.1.1: Identify Type Of Differential Equations Order в†’ The number of the highest derivative in a differential equation. Separable differential equations can be described as first-order first-degree differential equations where the expression for the derivative in terms of the variables is a multiplicatively separable function of the two variables.

solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example. differential equation at the twelve points indicated. b) Let y f x be the particular solution to the differential equation with the initial condition f 1 1.

Section 2-3 : Exact Equations. The next type of first order differential equations that weвЂ™ll be looking at is exact differential equations. Before we get into the full details behind solving exact differential equations itвЂ™s probably best to work an example that will help to show us just what an exact differential equation вЂ¦ 25/08/2011В В· A basic lesson on how to solve separable differential equations. Such equations have important applications in the modelling of dynamic phenomena. Such equations have important applications in the

An example of a linear equation is because, for , it can be written in the form Notice that this differential equation is not separable because itвЂ™s impossible to factor the We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at вЂ¦

For similar discussion and examples, see David LomenвЂ™s article вЂњSolving Separable Differential Equations: Antidifferentiation and Domain Are Both NeededвЂќ in the Course Home Pages section of AP Calculus at the AP Central website. Answer interactive questions on separable differential equations. See what you know about specifics like how to solve a differential equations with 0 as a variable and how to identify a separable

View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦ Section 2-3 : Exact Equations. The next type of first order differential equations that weвЂ™ll be looking at is exact differential equations. Before we get into the full details behind solving exact differential equations itвЂ™s probably best to work an example that will help to show us just what an exact differential equation вЂ¦

We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at вЂ¦ PaulвЂ™s Online Notes, emphasizes this fact when stating that for a differential equation to be separable, all the yвЂ™s in the differential equation must be multiplied by the derivative, and all the xвЂ™s in the differential equation must be on the other side of the equal sign.

25/08/2011В В· A basic lesson on how to solve separable differential equations. Such equations have important applications in the modelling of dynamic phenomena. Such equations have important applications in the Section 2-3 : Exact Equations. The next type of first order differential equations that weвЂ™ll be looking at is exact differential equations. Before we get into the full details behind solving exact differential equations itвЂ™s probably best to work an example that will help to show us just what an exact differential equation вЂ¦

Examples Solve the (separable) differential equation Solve the (separable) differential equation Solve the following differential equation: Sketch the family of solution curves. Videos See short videos of worked problems for this section. Quiz. Take a quiz. Exercises See Exercises for 3.3 Separable Differential Equations (PDF). Work online to solve the exercises for this section, or for any Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria.

What is volcano sleep deprivation in high school students statistics translate a sentence into an equation and solve human resources current events i search paper ideas research approach and design teaching through problem solving pdf trigonometry word problems pdf with answers body language examples and meanings how to study for the bar exam in one month amway new diamonds 2016 вЂ¦ View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦

Separable Diп¬Ђerential Equations A diп¬Ђerential equation is an equation for an unknown function that involves the derivative of the unknown function. View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦

### Separable Differential Equations analyzemath.com

Differential Equations Equations Differential Calculus. Exact Differential Equations вЂў Integrating Factors Exact Differential Equations In Section 5.6, you studied applications of differential equations to growth and decay problems. In Section 5.7, you learned more about the basic ideas of differential equa-tions and studied the solution technique known as separation of variables. In this chapter, you will learn more about solving differential, solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example..

### Linear Differential Equations web.stanford.edu

Separable differential equation Calculus. Separable Diп¬Ђerential Equations A diп¬Ђerential equation is an equation for an unknown function that involves the derivative of the unknown function. https://en.wikipedia.org/wiki/Inseparable_differential_equation Separable equations are the class of differential equations that can be solved using this method. "Separation of variables" allows us to rewrite differential equations so we obtain an equality between two integrals we can evaluate..

Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria. Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria.

Examples Solve the (separable) differential equation Solve the (separable) differential equation Solve the following differential equation: Sketch the family of solution curves. Videos See short videos of worked problems for this section. Quiz. Take a quiz. Exercises See Exercises for 3.3 Separable Differential Equations (PDF). Work online to solve the exercises for this section, or for any What is volcano sleep deprivation in high school students statistics translate a sentence into an equation and solve human resources current events i search paper ideas research approach and design teaching through problem solving pdf trigonometry word problems pdf with answers body language examples and meanings how to study for the bar exam in one month amway new diamonds 2016 вЂ¦

Basics and Separable Solutions We now turn our attention to differential equations in which the вЂњunknown function to be deter- minedвЂќ вЂ” which we will usually denote by u вЂ¦ We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at вЂ¦

Mixing Tank Separable Differential Equations Examples When studying separable differential equations, one classic class of examples is the mixing tank problems. Here we will consider a few variations on this classic. Example 1. A tank has pure water п¬‚owing into it at 10 l/min. The contents of the tank are kept thoroughly mixed, and the contents п¬‚ow out at 10 l/min. Initially, the tank Exact Differential Equations вЂў Integrating Factors Exact Differential Equations In Section 5.6, you studied applications of differential equations to growth and decay problems. In Section 5.7, you learned more about the basic ideas of differential equa-tions and studied the solution technique known as separation of variables. In this chapter, you will learn more about solving differential

An example of a linear equation is because, for , it can be written in the form Notice that this differential equation is not separable because itвЂ™s impossible to factor the A п¬Ѓrst-order differential equation is called separable if it can be written in the form p(y) dy dx = q(x). (1.4.1) The solution technique for a separable differential equation is given in Theorem 1.4.2. Theorem 1.4.2 If p(y)and q(x)are continuous, then Equation (1.4.1) has the general solution p(y)dy= q(x)dx+c, (1.4.2) where c is an arbitrary constant. Proof

differential equation at the twelve points indicated. b) Let y f x be the particular solution to the differential equation with the initial condition f 1 1. solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example.

Separable differential equations can be described as first-order first-degree differential equations where the expression for the derivative in terms of the variables is a multiplicatively separable function of the two variables. A first order differential equation \(yвЂ™ = f\left( {x,y} \right)\) is called a separable equation if the function \(f\left( {x,y} \right)\) can be factored into the product of two functions of \(x\) and \(y:\)

We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at вЂ¦ View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦

differential equation at the twelve points indicated. b) Let y f x be the particular solution to the differential equation with the initial condition f 1 1. A first order differential equation \(yвЂ™ = f\left( {x,y} \right)\) is called a separable equation if the function \(f\left( {x,y} \right)\) can be factored into the product of two functions of \(x\) and \(y:\)

A п¬Ѓrst-order differential equation is called separable if it can be written in the form p(y) dy dx = q(x). (1.4.1) The solution technique for a separable differential equation is given in Theorem 1.4.2. Theorem 1.4.2 If p(y)and q(x)are continuous, then Equation (1.4.1) has the general solution p(y)dy= q(x)dx+c, (1.4.2) where c is an arbitrary constant. Proof The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.

Separable Differential Equations We start with the deп¬Ѓnition of a separable diп¬Ђerential equation. Deп¬Ѓnition 1.1. A separable equation is a п¬Ѓrst order diп¬Ђerential equa- tion in which the expression for dy/dx can be factcored as a function of x times a function of y. In other words, it is an equation of the from dy dx = g(x) f(y) (we write it as a fraction for convenience). To solve Separable Differential Equation. Sanjay is a microbiologist, and he's trying to come up with a mathematical model to describe the population growth of a certain type of bacteria.

A separable differential equation is a common kind of differential equation that is especially straightforward to solve. Separable equations have the form \(\frac{dy}{dx}=f(x)g(y)\), and are called separable because the variables \(x\) and \(y\) can be brought to opposite sides of the equation. View, download and print Worksheet 5.1 - Separable Differential Equations With Answers - Calculus Maximus pdf template or form online. 392 Equation Worksheet вЂ¦

Mixing Tank Separable Differential Equations Examples When studying separable differential equations, one classic class of examples is the mixing tank problems. Here we will consider a few variations on this classic. Example 1. A tank has pure water п¬‚owing into it at 10 l/min. The contents of the tank are kept thoroughly mixed, and the contents п¬‚ow out at 10 l/min. Initially, the tank Answer interactive questions on separable differential equations. See what you know about specifics like how to solve a differential equations with 0 as a variable and how to identify a separable

Example 1: Solve and find a general solution to the differential equation. y ' = 3 e y x 2 Solution to Example 1: We first rewrite the given equations in differential form and with variables separated, the y's on one side and the x's on the other side as follows. solved separable differential equations. Such equations arise when investigating exponen- Such equations arise when investigating exponen- tial growth or decay, for example.

Differential equations arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in вЂ¦ DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS 1. Find the solution of y0 +2xy= x,withy(0) = в€’2. This is a linear equation. The integrating factor is e

What is volcano sleep deprivation in high school students statistics translate a sentence into an equation and solve human resources current events i search paper ideas research approach and design teaching through problem solving pdf trigonometry word problems pdf with answers body language examples and meanings how to study for the bar exam in one month amway new diamonds 2016 вЂ¦ What is volcano sleep deprivation in high school students statistics translate a sentence into an equation and solve human resources current events i search paper ideas research approach and design teaching through problem solving pdf trigonometry word problems pdf with answers body language examples and meanings how to study for the bar exam in one month amway new diamonds 2016 вЂ¦

25/08/2011В В· A basic lesson on how to solve separable differential equations. Such equations have important applications in the modelling of dynamic phenomena. Such equations have important applications in the Differential equations arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in вЂ¦