Theorem 12.25 (Initial value Problem - IVP convolution method). The unique solution to the initial value problem  

            ,  

with    and   ,  is given by  

            ,  

where    is the solution to the homogeneous equation  

            ,  

with  ,  and  

              has the Laplace transform given by  .  

Proof.

Proof of Theorem 12.25.

Proof of Theorem 12.25 is in the book.

Complex Analysis for Mathematics and Engineering

(c) 2006 John H. Mathews, Russell W. Howell