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Solve the initial value problems?

Solve the initial value problems.

(a) y'=y+e^(t/4); y(0)=2

This is first-order linear when re-written as the following:

y'-y=e^(t/4)

Integration factor=e^(∫-1 dt)=e^(-t)

Multiply both sides by integration factor.

e^(-t)y'-e^(-t)y=e^(-3t/4)

d/dt[e^(-t)*y]=e^(-3t/4)

Integrate both sides.

e^(-t)*y=-4/3*e^(-3t/4) + C

y=-4/3*e^(t/4) + C*e^(t)

2=-4/3+C

C=10/3

y=-4/3*e^(t/4)+10/3*e^(t)

(b)

Another first order linear problem

y'+2y=t*e^(-2t)

Integration factor=e^(∫ 2 dt)=e^(2t)

Multiply both sides by integration factor.

y'(e^(2t))+2*e^(2t)y=t

d/dt[y*e^(2t)]=t

y*e^(2t)=t^2/2 + C

y=t^2/2*e^(-2t)+C*e^(-2t)

0=1/2+C

C=-1/2

y=t^2/2*e^(-2t)-1/2*e^(-2t)