ODE Chapter 1: First Order Differential Equations using Separation of Variables

First Order Differential Equations using Separation of Variables

Below are solved examples of first order differential equations using the separation of variables method.

Example 1: Solve \( \frac{dy}{dx}=3x^2 \)

View Solution

Separate variables \[ dy = 3x^2 dx \] Integrate both sides \[ \int dy = \int 3x^2 dx \] \[ y = x^3 + C \]

Example 2: Solve \( \frac{dy}{dx}=xy \)

View Solution

Separate variables \[ \frac{dy}{y}=x\,dx \] Integrate \[ \int \frac{1}{y}dy = \int x\,dx \] \[ \ln |y| = \frac{x^2}{2}+C \] \[ y = Ce^{x^2/2} \]

Example 3: Solve \( \frac{dy}{dx}=\frac{x}{y} \)

View Solution

Separate variables \[ y\,dy = x\,dx \] Integrate \[ \int y\,dy = \int x\,dx \] \[ \frac{y^2}{2} = \frac{x^2}{2} + C \] \[ y^2 = x^2 + C \]

Example 4: Solve \( \frac{dy}{dx}=y^2 \)

View Solution

Separate variables \[ \frac{dy}{y^2}=dx \] Integrate \[ \int y^{-2}dy = \int dx \] \[ -\frac{1}{y}=x+C \] \[ y=\frac{1}{C-x} \]

Example 5: Solve \( \frac{dy}{dx}=\frac{1+y^2}{x} \)

View Solution

Separate variables \[ \frac{dy}{1+y^2}=\frac{dx}{x} \] Integrate \[ \int \frac{dy}{1+y^2}=\int \frac{dx}{x} \] \[ \tan^{-1}y=\ln |x|+C \] \[ y=\tan(\ln |x|+C) \]

Example 6: Solve \( \frac{dy}{dx}=y\cos x \)

View Solution

Separate variables \[ \frac{dy}{y}=\cos x\,dx \] Integrate \[ \int \frac{1}{y}dy=\int \cos x\,dx \] \[ \ln |y|=\sin x + C \] \[ y=Ce^{\sin x} \]

Example 7: Solve \( \frac{dy}{dx}=\frac{y}{1+x} \)

View Solution

Separate variables \[ \frac{dy}{y}=\frac{dx}{1+x} \] Integrate \[ \int \frac{1}{y}dy=\int \frac{1}{1+x}dx \] \[ \ln |y|=\ln |1+x|+C \] \[ y=C(1+x) \]

Example 8: Solve \( \frac{dy}{dx}=xe^{y} \)

View Solution

Separate variables \[ e^{-y}dy=x\,dx \] Integrate \[ \int e^{-y}dy=\int x\,dx \] \[ -e^{-y}=\frac{x^2}{2}+C \]

Example 9: Solve \( \frac{dy}{dx}=\frac{x^2}{y+1} \)

View Solution

Separate variables \[ (y+1)dy=x^2dx \] Integrate \[ \int (y+1)dy=\int x^2dx \] \[ \frac{y^2}{2}+y=\frac{x^3}{3}+C \]

Example 10: Solve \( \frac{dy}{dx}=y(1+x) \)

View Solution

Separate variables \[ \frac{dy}{y}=(1+x)dx \] Integrate \[ \int \frac{1}{y}dy=\int (1+x)dx \] \[ \ln |y|=x+\frac{x^2}{2}+C \] \[ y=Ce^{x+x^2/2} \]