Solving Ordinary Differential Equations Using Separable Equations: Techniques and Examples

Separable ordinary differential equations (ODEs) are a class of differential equations that can be solved by separating the variables and integrating. This article explores the step-by-step process of solving such equations, provides an example, and discusses the broader context of separability in the field of differential equations.

Introduction to Separable ODEs

A separable ordinary differential equation (ODE) is one that can be written in the form:

[frac{dy}{dx} g(x)h(y)]" separator"">

Where [g(x)] is a function of [x] and [h(y)] is a function of [y].

Step-by-Step Solution of Separable ODEs

To solve a separable ODE, we follow these steps:

Step 1: Write the ODE in the Standard Form

Ensure that the ODE is in the form:

[frac{dy}{dx} g(x)h(y)]

This is the standard form of a separable ODE.

Step 2: Separate the Variables

Rearrange the equation to separate the variables [y] and [x]:

[frac{1}{h(y)} :dy g(x) :dx]

Step 3: Integrate Both Sides

Integrate both sides of the equation:

[int frac{1}{h(y)} :dy int g(x) :dx]

Step 4: Solve the Integrals

This will typically yield a function of [y] on one side and a function of [x] on the other side plus a constant of integration [C]:

[F(y) G(x) C]

Where [F(y)] is the antiderivative of [frac{1}{h(y)}], and [G(x)] is the antiderivative of [g(x)].

Step 5: Solve for [y] if possible

Usually, depending on the form of the functions, it is possible to solve for [y] in terms of [x].

Step 6: Apply Initial Conditions if any

If initial conditions are provided, substitute them into the equation to solve for the constant [C].

Example: A Simple Separable ODE

Consider the ODE:

[frac{dy}{dx} y sin x]

Separate Variables:

[frac{1}{y} :dy sin x :dx]

Integrate Both Sides:

[int frac{1}{y} :dy int sin x :dx]

Integrate:

[ln |y| -cos x C]

Exponentiate to Solve for [y]:

[|y| e^{-cos x C}]

Let [K e^C] so:

[y K e^{-cos x}]

Initial Conditions: If an initial condition is given, say [y(x_0) y_0], substitute to find [K].

Broad Context of Separability in Differential Equations

References to separability in the field of differential equations can sometimes lead to confusion, as it is also used in the context of solving certain types of partial differential equations (PDEs). For instance, consider the PDE:

[ abla^2 phi(x,y) - lambda^2 phi(x,y) 0]

Where [phi(x,y) mu(x) u(y)]. This gives:

[mu''(x) u(y) - lambda^2 mu(x) u(y) 0]

Dividing through by [mu u] yields:

[frac{mu''(x)}{mu(x)} - lambda^2 - frac{ u''(y)}{ u(y)} gamma^2]

Since both expressions are equal to the third, we have:

[ u''(y) - gamma^2 u(y) 0]

[mu''(x) - (lambda^2 - gamma^2) mu(x) 0]

Our general solution is of the form:

[phi(x,y) mu(x) u(y) (A cos gamma y B sin gamma y)(C cos sqrt{lambda^2 - gamma^2} x D sin sqrt{lambda^2 - gamma^2} x)]

Boundary conditions can give us particular values of the constants.

This broad context of separability in PDEs is important to understand when dealing with more complex problems in physics and engineering.