# Solve Simple Differential Equations

This is a tutorial on solving simple first order differential equations of the form  $\dfrac{dy}{dx} = f(x)$ A set of examples with detailed solutions is presented and a set of exercises is presented after the tutorials. Depending on $$f(x)$$, these equations may be solved analytically by integration. In what follows $$C$$ is a constant of integration and can take any constant value.

## Examples with Solutions

### Example 1:

Solve and find a general solution to the differential equation. $\dfrac{dy}{dx} = 2x + 1$ Solution to Example 1:
Integrate both sides of the equation. $\int y' \, dx = \int (2x + 1) \, dx$ which gives $y = x^2 + x + C$ As a practice, verify that the solution obtained satisfy the differential equation given above.

### Example 2:

Solve and find a general solution to the differential equation. $2\dfrac{dy}{dx} = \sin(2x)$ Solution to Example 2:
Write the differential equation of the form $$y' = f(x)$$. $y' = \dfrac{1}{2} \sin(2x)$ Integrate both sides $\int y' \, dx = \int \dfrac{1}{2} \sin(2x) \, dx$ Let $$u = 2x$$ so that $$du = 2 dx$$, the right side becomes $y = \int \dfrac{1}{4} \sin(u) \, du$ Which gives $y = -\dfrac{1}{4} \cos(u) = -\dfrac{1}{4} \cos(2x)$

### Example 3:

Solve and find a general solution to the differential equation.
$y' e^{-x} + e^{2x} = 0$
Solution to Example 3:
Multiply all terms of the equation by $$e^x$$, simplify and write the differential equation of the form $$y' = f(x)$$. $y' = -e^{3x}$ Integrate both sides of the equation $\int y' \, dx = \int -e^{3x} \, dx$ Let $$u = 3x$$ so that $$du = 3 dx$$, write the right side in terms of $$u$$ $y = \int -\dfrac{1}{3} e^{u} \, du$
Which gives $y = -\dfrac{1}{3} e^{u} = -\dfrac{1}{3} e^{3x}$

## Exercises

Solve the following differential equations.
a) $$2\dfrac{dy}{dx} = 6x$$
b) $$y' \cos(x) = \sin(2x)$$
c) $$y' e^{x} = e^{3x}$$

Solutions to the above exercises
a) $$y = \dfrac{3}{2} x^2 + C$$
b) $$y = -2 \cos(x) + C$$
c) $$y = \dfrac{1}{2} e^{2x} + C$$