Find the solution of the given initial value problem:
(a) y' + 2y = te^{-2t}, y(1) = 0
(b) t^{3}y' + 4t^{2}y = e^{-t}, y(-1) = 0

Step-by-step explanation:

(a)

To find the complementary function

   D+2 = 0

=> D = -2

So, the complementary function can by given by

Now, to find particular integral

               = not defined

So,

So, complete solution can be given by

As given in question

Hence, the complete solution can be give by

(b)

Integrating factor can be given by

Now , the solution of the given differential equation can be given by

According to question

Now, the complete solution of the given differential equation cab be given by

a.

b.

Step-by-step explanation:

We are given that

a.

Compare with

We have P(t)=2,Q(t)=

Integration factor=

Substitute the values then, we get

Substitute the values x=1 and y=0

Then, we get

Substitute the value in the given function

Hence,

b.

I.F=

Substitute the values x=-1,y=0 then, we get

C=0

Substitute the value of C then we get

the answer is 62