# Solving Differential Equations by Separation of Variables

### Solving Differential Equations by Separation of Variables:

1. Method Overview:

• Applicable to certain first-order ordinary differential equations (ODEs) that can be written in the form $\frac{dy}{dx}=f\left(x\right)g\left(y\right)$ or a similar separable form.
• Involves isolating variables (putting all $y$ terms on one side and all $x$ terms on the other) before integrating both sides separately.
2. Steps to Solve by Separation of Variables:

a. Identify a Separable Form: Ensure the differential equation can be written in the form $\frac{dy}{dx}=f\left(x\right)g\left(y\right)$ or similar.

b. Isolate Variables: Rearrange the equation to have all $y$ terms on one side and all $x$ terms on the other side.

c. Integrate Both Sides: Integrate both sides of the equation with respect to their respective variables.

d. Add Constant of Integration: Include a constant of integration when integrating each side of the equation.

3. Example: Solving a Differential Equation by Separation of Variables:

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

a. Separate Variables: Rewrite the equation by isolating variables: $\frac{dy}{y}={x}^{2}\text{\hspace{0.17em}}dx$

b. Integrate Both Sides:

• Integrate the left side with respect to $y$ and the right side with respect to $x$: $\int \frac{1}{y}\text{\hspace{0.17em}}dy=\int {x}^{2}\text{\hspace{0.17em}}dx$
• This yields $\mathrm{ln}\mathrm{\mid }y\mathrm{\mid }=\frac{{x}^{3}}{3}+{C}_{1}$, where ${C}_{1}$ is the constant of integration.

c. Solve for $y$:

• Exponentiate both sides to solve for $y$: $\mathrm{\mid }y\mathrm{\mid }={e}^{\frac{{x}^{3}}{3}+{C}_{1}}={e}^{{C}_{1}}\cdot {e}^{\frac{{x}^{3}}{3}}$
• $\mathrm{\mid }y\mathrm{\mid }=C{e}^{\frac{{x}^{3}}{3}}$ where $C=±{e}^{{C}_{1}}$.

d. Final Result:

• The solution to the differential equation $\frac{dy}{dx}={x}^{2}y$ by separation of variables is $y=C{e}^{\frac{{x}^{3}}{3}}$, where $C$ is an arbitrary constant.