Solution of General Form of Variable Separation

The method of separation of variables is a technique used to solve ordinary differential equations (ODEs) where the variables can be separated into independent parts. This method is particularly useful for solving first-order differential equations of the form:

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

The idea is to rearrange the equation so that each side contains only one variable and its differential. Then, we integrate both sides to find the solution.

Steps to Solve Using Separation of Variables

1. Separate the Variables: Rewrite the equation in the form where all terms involving $y$ are on one side and all terms involving $x$ are on the other side:

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

1. Integrate Both Sides: Integrate the left side with respect to $y$ and the right side with respect to $x$:

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

1. Solve for $y$: If possible, solve the resulting equation for $y$ to get the explicit solution. If not, you may leave the solution in implicit form.

Example

Solve the differential equation:

$$ \frac{dy}{dx} = \frac{x}{y} $$

Step 1: Separate the Variables

$$ y \cdot dy = x \cdot dx $$

Step 2: Integrate Both Sides

$$ \int y \cdot dy = \int x \cdot dx $$ $$ \frac{y^2}{2} = \frac{x^2}{2} + C $$

Step 3: Solve for $y$

$$ y^2 = x^2 + C' $$ $$ y = \pm \sqrt{x^2 + C'} $$

Here, $C'$ is a constant that represents $2C$.

Important Points and Differences

Aspect	Description
Applicability	Only applicable to ODEs that can be separated into a product of functions of $x$ and $y$.
Solution Form	Can result in either explicit or implicit solutions.
Integration	Requires integration of both sides, which may involve indefinite or definite integrals.
Initial Conditions	If given, can be used to find the particular solution by determining the constant of integration.
Difficulty	The difficulty of integration can vary greatly depending on the functions $g(x)$ and $h(y)$.

Formulas

The general form of a separable differential equation and its separated form are given by:

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

After integrating, we get:

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

Additional Example

Solve the differential equation with an initial condition:

$$ \frac{dy}{dx} = 3x^2y, \quad y(1) = 2 $$

Step 1: Separate the Variables

$$ \frac{1}{y}dy = 3x^2dx $$

Step 2: Integrate Both Sides

$$ \int \frac{1}{y}dy = \int 3x^2dx $$ $$ \ln|y| = x^3 + C $$

Step 3: Solve for $y$

$$ y = e^{x^3 + C} $$

Step 4: Apply Initial Condition

$$ y(1) = e^{1^3 + C} = 2 $$ $$ e^C = 2 $$ $$ C = \ln(2) $$

Final Solution

$$ y = e^{x^3 + \ln(2)} $$ $$ y = 2e^{x^3} $$

In conclusion, the method of separation of variables is a powerful tool for solving certain types of differential equations. It is important to recognize when a differential equation is separable and to be comfortable with the integration techniques required to find the solution.