Solving Differential Equations by Separation of Variables
Solving Differential Equations by Separation of Variables:

Method Overview:
 Applicable to certain firstorder ordinary differential equations (ODEs) that can be written in the form $\frac{dy}{dx}=f(x)g(y)$ 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.

Steps to Solve by Separation of Variables:
a. Identify a Separable Form: Ensure the differential equation can be written in the form $\frac{dy}{dx}=f(x)g(y)$ 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.

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 $C=\pm {e}^{{C}_{1}}$.
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.