This is a question in electrical book but really hard to sol

This is a question in electrical book but really hard to solve by simple methods ):

please help me with clear steps especailly for part b ( problem 1 part b)

Thank you very much

Problem 1: Solve the following differential or difference equations, with the specified initial condition: Problem 1: Solve the following differential or difference dy(t)

Solution

a.

Rewrite it as

dy/dt+2/3 y=2 e^{t/3 }

Integrating factor is

e^{2t/3}

Multiplying by this integrating factor gives

(dy/dt+2y/3)e^{2t/3}=2e^t

d/dt(y e^{2t/3})=2e^t

Integrating gives

y e^{2t/3}=2e^t+C

y=2e^{t/3}+Ce^{-2t/3}

y(0)=2+C=3 giving C=1

b.

First we solve homogeneous equatoin

2y[n]-y[n-1]=0

THis is a linear homogeneous recurrence with constant coefficeints

So solution is of the form: y[n]=r^n

Substituting gives

2r-1=0

r=1/2

So, y[n]=A/2^n

Now for the inhomogeneous we need to find one particular solution

Let particular solution be :yp[n]=B2^n

Substituting gives

2*B2^n-2^{n-1}=2^n

2B-1/2=1

B=3/4

So, yp=3*2^{n-2}

SO,

y[n]=A/2^n+3*2^{n-2}

y[0]=3=A+3/4

A=9/4

y[n]=9/2^{n+2}+3*2^{n-2}