Page 127 - Schaum's Outline of Differential Equations

CHAPTER 13

Initial-Value

Problems for Linear

Differential Equations

[nilial-\alue problems are solved h\ appKing ihc initial conditions lo ihc general solution of [he differential
equation, ll nuisl he emphasi/ed !hal ihc initial conditions are applied aiilv lo Ihc general solution and mil to the
homogeneous s{ilulion y,,. even though il is y/, lhai possesses all ihe arbitrary consianls lhai must be evaluated.
The one exception is when ihc general solution is ihe homogeneous solution; lhai is, when the dilTerenlial equation
under eon si derail on is iise.ll' homogeneous.

Solved Problems

13.1. Solve y" - y' - 2v = 4\-; v(0) = 1. v'(0) = 4.
The jicneral solution of the differential equation is given in Problem 1 . I as
I

Therefore.

Applying ihc first initial condition to (/). we obtain

Applying the second initial condition to (2). we obtain

Solving (3) and (4) simultaneously, we find lhai c t = 2 and c 2 = 2. Substituting these values into (/). ne obtain ihc
solution of I he initial-value problem as
y=2e+2e-2x+2x-3

111
Copyright © 2006, 1994, 1973 by The McGraw-Hill Companies, Inc. Click here for terms of use.