# Solve Simple Differential Equations

This is a tutorial on solving simple first order differential equations of the form

## Examples

Example 1: Solve and find a general solution to the differential equation.

y ' = 2x + 1

__Solution to Example 1:__

Integrate both sides of the equation.

ò y ' dx = ò (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 y ' = sin(2x)

__Solution to Example 2:__

Write the differential equation of the form y ' = f(x).

y ' = (1/2) sin(2x)

Integrate both sides

ò y ' dx = ò (1/2) sin(2x) dx

Let u = 2x so that du = 2 dx, the right side becomes

y = ò (1/4) sin(u) du

Which gives.

y = (-1/4) cos(u) = (-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} and write the differential equation of the form y ' = f(x).

y ' = - e^{ 3x}

Integrate both sides of the equation

ò y ' dx = ò - e^{ 3x}
dx

Let u = 3x so that du = 3 dx, write the right side in terms of u

y = ò (-1/3) e^{ u} du

Which gives.

y = (-1/3) e^{ u} = (-1/3) e^{ 3x}

## Exercises

Solve the following differential equations.a) 2y ' = 6x

b) y ' cos x = sin(2x)

c) y ' e

^{ x}= e

^{ 3x}

__Solutions to the above exercises__

a) y = (3/2) x

^{ 2}+ C

b) y = -2 cos x + C

c) y =(1 / 2) e

^{ 2x}+ C

More references on
Differential Equations

Differential Equations - Runge Kutta Method