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Math A564 Operest sal Methods for sogicbers “Review of ODD E's: Once the Laplace Transform is imtroauced, some of the First problems to be solveal Using the transform are linear constant coefficient odes, We review solution techniques used in the differential equations SooUrse, . First order. linear equation S: , -t Example: Solve the initla/ value problem: yy? e441, yo=3 o) Tntesr ing Factor: Ho Hiplying bby creates 3 new lePt side that is the derivative ofa product, ‘ Syrely = (etyy'= ive o ey =teeec Le —t. yo = tet £+C@. Impose the initial condition (ehveys the lest step) ytors Lc 3.2 C2 ana v(t) tehdeze® y= = (120€ ad b) Compile mentary & Particular Solvtions: Sjhce the equation is linear ana nonhomogeneous, we can express he enersl solution as the sum ofa complementary solution anc a par Icy lar solution, Los -t Complementary aelvbion: yt Y. FO y= Ce (Particular Solution: Using the Method of Undetermined Coe FFicients, noting that tiga solution of the homagensous equation: Ny “Hey =e 4 io, where we assume: yrAte Lp PoP . Pp a Ys AS Ate and yey = AS AE NESE =e Equating coePicients: Ae4,B+1 arc ye bent Gereral solution: chs YGQ)+ a) = Ce te teh vereralseivtion: y= Yer : nd -Em posing the initlal condition we again oblain: y= Gr2)e +4. Remark: Wecan view the solution as the sum of a transient tecrn, Ctraje™*, which approaches O as b* 00 and 2 steady state term, 4, which persists. 
