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Chapter2 SolvingFirst-Order Equations
from SOLUTIONS MANUAL for Differential Equations Theory, Technique, and Practice 3rd Edition By Steven G
by ACADEMIAMILL
2.1SeparableEquations
1. Usethemethodofseparationofvariablestosolveeachoftheseordinary differentialequations.
(a)Writetheequation x5y + y5 =0inLeibnitzform x5 dy dx + y5 =0 andseparatethevariables: dy y5 = dx x5 .Integrate, dy y5 = dx x5 ,to obtainthesolution: y 4/( 4)= x 4/4+C.Thiscanalsobewritten intheform x
(b)Writetheequation y =4xy inLeibnitzform dy dx =4xy andseparate thevariables: dy y =4xdx.Integrate, dy y = 4xdx,toobtainthe solution:ln |y| =2x2 + C.Thiscanalsobewrittenintheform y = Ce2x 2
(c)Writetheequation y + y tan x =0inLeibnitzform dy dx + y tan x =0 andseparatethevariables: dy y = tan xdx.Integrate, dy y == tan xdx,toobtainthesolution:ln |y| =ln | cos x| + C.Thiscan alsobewrittenintheform y = C cos x.
(d)Theequation(1+ x2)dy +(1+ y2)dx =0canberearrangedand integrateddirectly, dy 1+y2 + dx 1+x2 = C.Therefore,theimplicit solutionisarctan y +arctan x = C.Thiscanalsobewritteninthe form y =tan(C arctan x).
(e)Proceedasinpart(d).Rearrange y ln ydx xdy =0totheform dx x dy y ln y =0andintegrate: dx x dy y ln y = C.Thisyieldsthe implicitsolutionln |x|− ln | ln y| = C whichcanalsobewrittenin theformln y = Cx or y = eCx .
