chapter menu

8. Using Inverse Laplace Transforms to Solve Differential Equations

Laplace Transform of Derivatives

We use the following notation:

Later, on this page...

Subsidiary Equation


(a) If we have the function `g(t)`, then `G(s) = G = ` ℒ`{g(t)}`.

(b) g(0) is the value of the function g(t) at t = 0.

(c) g'(0), g’’(0),... are the values of the derivatives of the function at t = 0.

If `g(t)` is continuous and g'(0), g’’(0),... are finite, then we have the following.

First Derivative

ℒ`{g"'"(t)}=` ℒ`{(dg)/(dt)}` `=sG-g(0)`

Second Derivative

ℒ`{g"''"(t)}=s^2G-s\ g(0) - g"'"(0)`

We saw many of these expressions in the Table of Laplace Transforms.

If instead of g(t) we have a function y of x, then Equation (2) would simply become:

ℒ`{y’’(x)} = s^2Y − s\ y (0) − y’(0)`

Likewise, if we have an expression for current i and it is a function of t, then the equation would become:

ℒ`{i’’(t)} = s^2I − s\ i(0) − i\ ’(0)`

For the n-th derivative

ℒ`{(d^ng)/(dt^n)}=s^nG-s^(n-1)g(0)-s^(n-2)g ’ (0)-...-g^(n-1)(0)`

NOTATION NOTE: If we have y and it is a function of t, then the notation would become:

ℒ`{(d^ny)/(dt^n)}=s^nY-s^(n-1)y(0)-s^(n-2)y ’ (0)-...-y^(n-1)(0)`

Subsidiary Equation

The subsidiary equation is the equation in terms of s, G and the coefficients g'(0), g’’(0),... etc., obtained by taking the transforms of all the terms in a linear differential equation.

The subsidiary equation is expressed in the form G = G(s).


Write down the subsidiary equations for the following differential equations and hence solve them.

Example 1

`(dy)/(dt)+y=sin\ 3t`, given that y = 0 when t = 0.

Solution Graph for Example 1

This is the graph of the solution we obtained in the example above.


Example 2

Solve `(d^2y)/(dt^2)+2(dy)/(dt)+5y=0`, given that `y = 1`, and `(dy)/(dt)=0`, when `t = 0`.

Solution Graph for Example 2

Here is the graph of what we just found:


Example 3

`(d^2y)/(dt^2)-2(dy)/(dt)+y=e^t`, given that y = -2, and `(dy)/(dt)=-3` when t = 0.

Solution Graph for Example 3



The current i(t) in an electrical circuit is given by the DE

`(d^2i)/(dt^2)+2(di)/(dt)=0,if 0 < t < 10`

`=1,if 10 < t < 20`

`=0,if t > 20`

and i(0) = 0, i’(0) = 0.

Determine the current as a function of t.

Solution Graph for the Application (current at time t)


Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Calculus Lessons on DVD


Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.