Differential Equations and Linear Algebra, 4th Edition

10.2 Transformation of Initial Value Problems

We now discuss the application of Laplace transforms to solve a linear differential equation with constant coefficients, such as

$$ax″(t)+bx\text{′}(t)+cx(t)=f(t),$$ (1)

with given initial conditions $x(0)=x_0$ and $x^{\text{′}}(0)={x^{\text{′}}}_0\text{.}$ By the linearity of the Laplace transformation, we can transform Eq. (1) by separately taking the Laplace transform of each term in the equation. The transformed equation is

$$a\mathcal{L}\{x″(t)\}+b\mathcal{L}\{x\text{′}(t)\}+c\mathcal{L}\{x(t)\}=\mathcal{L}\{f(t)\};$$ (2)

it involves the transforms of the derivatives $x\text{′}$ and $x″$ of the unknown function x(t). The key to the method is Theorem 1, which tells us how to express the transform of the derivative of a function in terms of the transform of the function itself.

THEOREM 1 Transforms of Derivatives ...