Question

1. (4 points) Determine whether the given function y, given explicit or implicit, is a solution to the corresponding differential equation a) y = 2* +3e2a; y" - 3y + 2y = 0. dy 2.ry b) y - In y = r2+1, (Use implicit differentiation) dr y-1 2. (3 points) Find the solution to the initial value problem: dy = e(t+1); y(2) = 0 dr 3. (3 points) Find the general solution to the following equation. y dy ada COS

Answer 1

@ y=20% +30 an we have do cheones y is a Solution of on nood. yl-3y + 2y = 0 y=204+302n - y = 200 +6eon y"=2 et + 12 gan Pulling the value as I th' and y" in letis of the above -+ts = /0* +12,62% - 669-18** + 4634 gek Since y daliskied died so g is a Sot is the di-en; We have 0 Q. Hos di ed gl-sy't she

y-lny axat by implicit dess; we have => y .y = (1-3) =21 dy dx dy 2x 드 dy da axy yol qxy (1-by) y-T which is giver en dy da any So y-lay = x + is a set of dy x xth da -y x ; = &- y (x+1) ; y(2)=0 dy = Integrating both dides We have > Jedy = ((1+z) du - & - atlanto Put y(2)=0 => 0 = &+ lina tc => c= -|- In 2 > C=-(lol+la 2) -lna

So She solh is y = lox tx-lna lo a tla a = x >/09 = x + ln (42)

3 >> dy dx = Cos x y dy = kloste du Integrating bosh she dides we have Sydy=Sx Cose du dx y2 & gq x. I cosa de - SS Cosx dx ). d(x) asina- salinx. I de - a dinx + cosa to y2 = 2 (usinx & Cosat