Solution of Differential Equations by Variable Separable Method

Differential equations (DEs) are equations that involve derivatives of one or more functions. The variable separable method is one of the simplest techniques for solving certain types of ordinary differential equations (ODEs). This method is applicable when the equation can be written in a form where the variables can be separated on different sides of the equation.

Understanding Variable Separable Method

A first-order ODE is said to be separable if it can be expressed in the form:

$$ \frac{dy}{dx} = g(x)h(y) $$

The idea is to rearrange the equation so that each variable appears with its own derivative:

$$ \frac{1}{h(y)}dy = g(x)dx $$

Then, we integrate both sides to find the solution:

$$ \int \frac{1}{h(y)}dy = \int g(x)dx $$

Steps to Solve DE Using Variable Separable Method

1. Write the differential equation in the form $\frac{dy}{dx} = g(x)h(y)$.
2. Rearrange the equation to separate the variables: $\frac{1}{h(y)}dy = g(x)dx$.
3. Integrate both sides of the equation.
4. Solve for $y$ if possible.
5. Include the constant of integration.

Table of Differences and Important Points

Aspect	Variable Separable Method	Other Methods (e.g., Linear, Exact)
Applicability	Only for separable ODEs	For various types of ODEs
Complexity	Generally simpler	Can be more complex
Integration	Requires direct integration	May involve integrating factors or more advanced techniques
Solution	Explicit solution is often possible	Solution may be implicit or require numerical methods
Initial Conditions	Can be applied after integration	Often applied during the solving process

Formulas

• Separable ODE: $\frac{dy}{dx} = g(x)h(y)$
• Separated form: $\frac{1}{h(y)}dy = g(x)dx$
• Integrated form: $\int \frac{1}{h(y)}dy = \int g(x)dx + C$

Examples

Example 1: Basic Separable ODE

Solve the differential equation $\frac{dy}{dx} = xy$.

Solution:

1. Separate the variables: $\frac{1}{y}dy = xdx$.
2. Integrate both sides: $\int \frac{1}{y}dy = \int xdx$.
3. This gives $\ln|y| = \frac{1}{2}x^2 + C$.
4. Solve for $y$: $y = \pm e^{\frac{1}{2}x^2 + C} = Ce^{\frac{1}{2}x^2}$, where $C$ is the constant of integration.

Example 2: Applying Initial Conditions

Solve the differential equation $\frac{dy}{dx} = 3y$ with the initial condition $y(0) = 2$.

Solution:

1. Separate the variables: $\frac{1}{y}dy = 3dx$.
2. Integrate both sides: $\int \frac{1}{y}dy = \int 3dx$.
3. This gives $\ln|y| = 3x + C$.
4. Apply the initial condition: $\ln|2| = 3(0) + C$ gives $C = \ln(2)$.
5. Solve for $y$: $y = e^{3x + \ln(2)} = 2e^{3x}$.

Example 3: Nonlinear Separable ODE

Solve the differential equation $\frac{dy}{dx} = \frac{x}{y^2}$.

Solution:

1. Separate the variables: $y^2dy = xdx$.
2. Integrate both sides: $\int y^2dy = \int xdx$.
3. This gives $\frac{1}{3}y^3 = \frac{1}{2}x^2 + C$.
4. Solve for $y$: $y = \left( \frac{3}{2}x^2 + 3C \right)^{\frac{1}{3}}$.

In conclusion, the variable separable method is a powerful tool for solving first-order ODEs when the equation can be written in a form that allows for the separation of variables. It involves straightforward integration and often yields an explicit solution. However, it is limited to equations that can be manipulated into the separable form.