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

y ' = 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.
__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