## Monday Math 114

The Laplace Transform
Part 15: Applying the Laplace Transform – Differential Equations

As was briefly noted in this part, the primary use of Laplace transforms is in solving linear, ordinary differential equations, as the transform turns the derivatives into algebraic operatons. For a more in-depth example, let us consider the initial value problem
, .
As this has constant coefficients, we could find the indicial equation, solve for its roots, and obtain the general solution. However, to find the specific solution with the above initial conditions would then require differentiating the general solution twice, and inserting values to obtain three linear equations for the three constants. With the Laplace transform, we will directly insert the values for y(0), y‘(0), and y”(0); and so, obtain the particular solution we want with less steps.

Taking the Laplace transform of our differential equation, and letting  to simplify our notation,

,
and so our solution is the inverse Laplace transform of the above.
Since , we can use partial fractions on the above to find that
,
and so we see that
,
and using from here that the Laplace transform of an exponential is
, and from here that the frequency-shifted sine and cosine formulas are

and
,
we see that we have
,
and we have our solution.