Separation of Variables

Separation of Variables is a method for solving first-order differential equations of the form:

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

The idea is simple:

Put all the y-terms on one side and all the x-terms on the other,
then integrate both sides.

This method gives a general solution and often leads to exponential, logarithmic, or trigonometric functions depending on the equation.

General Steps for Separation of Variables

1. Start with the differential equation

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

2. Separate variables

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

3. Integrate both sides

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

4. Add the constant of integration (usually on the x-side)

5. Solve for y if possible (This creates the explicit solution.)

This method is widely used in growth/decay models, cooling laws, logistic equations, and many AP exam questions.

Example 1 (Simple Exponential Growth)

Solve:

$$\frac{dy}{dx}=3y$$

Step 1: Separate

$$\frac{1}{y}\frac{dy}{dx}=3 \longrightarrow \frac{1}{y}dy=3dx$$

Step 2: Integrate

$$\int\frac{1}{y}dy=\int3dx$$

$$\ln|y|=3x+C$$

Step 3: Solve for y

Exponentiate: $$e^{\ln|y|)=e^{3x+C} \longrightarrow |y|=e^{3x+C}=e^{3x}\cdot{e^{c}}=C\cdot{e^{3x}}$$

$$e^{c}=constant$$

So:

$$y=Ce^{3x}$$​

Example 2 (Product Form)

Solve:

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

Step 1: Separate

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

Step 2: Integrate

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

$$-\frac{1}{y^{2}}=\frac{x^{3}}{3}+C$$

Step 3: Solve for y

$$-\frac{1}{y^{2}}=\frac{x^{3}}{3}+C$$

This is the general implicit form of the solution.

Example 3 (Logarithmic Form)

Solve:

$$\frac{dy}{dx}=\frac{x}{y}$$​

Step 1: Separate

$$y dy=x dx$$

Step 2: Integrate

$$\int y dy=\int x dx$$

$$\frac{1}{2}y^{2}=\frac{1}{2}x^{2}+C$$

Step 3: Solve

Multiply by 2:

$$y^{2}=x^{2}+C$$​

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

Initial Conditions problem

If a problem gives an initial value:

$$y(x_{0})=y_{0}$$​

you plug it in AFTER finding the general solution.

Example:

$$\frac{dy}{dx}=3y , y(0)=4$$

General solution:

$$y=Ce^{3x}$$

Use initial condition:

$$4=C\cdot{e^{0}} \longrightarrow C=4$$

$$y=4e^{3x}$$​