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Introduction to Differential Equations

What is a differential equation?

An equation that involves one or more derivatives of an unknown function is called a differential equation. The order of the highest derivative included in a differential equation defines the order of this equation.

Examples

  • y ' = 3x ,

    the order of the highest derivative is 1 (y ' ) so the order of this differential equation is 1.

  • y '' + y' + y = 3x ,

    the order of the highest derivative is 2 (y '' ) so the order of this differential equation is 2.

  • -2 y ''' + y'' + y 4 = 3x ,

    the order of the highest derivative is 3 (y ''' ) so the order of this differential equation is 3.

y = f(x) is a solution of a differential equation if the equation is satisfied upon substitution of y and its derivatives into the differential equation.

Example:

Verify that y = C*e 4x + e 3x, where c is a constant, is a solution to the differential equation

y ' - 4y = -e 3x


y ' is given by

y ' = 4C*e
4x + 3e 3x

We now substitute y ' and y into the left side of the equation and simplify

y ' - 4y = 4C*e
4x + 3e 3x - 4 (C*e 4x + e 3x)

= 4C*e
4x + 3e 3x - 4C*e 4x - 4e 3x

= 4C*e
4x - 4C*e 4x + e 3x (3 - 4)

= - e
3x

Which is equal to the left side of the given equation and therefore y = C*e
4x + e 3x is a solution to the differential equation y ' - 4y = -e 3x.

Most of the work on differential equations consists in solving these equations. For example to solve the following differential example

y ' = 2x

Let us integrate both sides of the given equation as follows

y ' dx = 2x dx

which gives

y + C1 = x
2 + C2

where C1 and C2 are constants of integration. The solution y of the above equation is given by: y = x
2 + C, where C = C2 - C1.

More references on Differential Equations

Differential Equations - Runge Kutta Method



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Updated: 2 April 2013

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