Page 266 - Schaum's Outline of Differential Equations

CHAPTER 25

Solutions of Linear

Systems by Laplace

Transforms

THE METHOD

Laplace transforms are useful for solving systems of linear differential equations; that is, sets of two or
more differential equations with an equal number of unknown functions. If all of the coefficients are constants,
then the method of solution is a straightforward generalization of the one given in Chapter 24. Laplace
transforms are taken of each differential equation in the system; the transforms of the unknown functions are
determined algebraically from the resulting set of simultaneous equations; inverse transforms for the unknown
functions are calculated with the help of Appendix A.

Solved Problems

25.1. Solve the following system for the unknown functions u(x) and v(x):

Denote !£{u(x)} and !£{v(x)} by U(s) and V(s), respectively. Taking Laplace transforms of both differential
equations, we obtain

or

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