Question

Find the solution to the following initial value problem

y' -y = t - sint + e^(2t); y(0) = 0

Answer 1

Given differential equation is

This is Linear Differential Equation

where

P = - 1

Q = t - sinn t + e^2t

The solution is given by:

Substituting the values, e get:

     (1)

By Integration by parts,

                            (2)

By integration by parts,

                      (3)

                        (4)

Sustituting (2), (3) & (4) in (1), we get the solution as:

                       

                                         (5)

Given:

At t = 0, y = 0

So, (5) becomes:

0 = 0 -1 - 1/2 +1+C

So,

C = 1/2

So, the solution of the intial value problem is given by substituting in (5) as follows:

             

            

i