Solve Simple Differential Equations

How to solve simple differential equations?

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



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Updated: 2 April 2013

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