Consider the initial value problem ty 3y cost4 with 1 Find

Consider the initial value problem ty\' + 3y = cos(t)4 with =1 Find the solution to the initial value problem.

Solution

y\'+3y/t=cos(t^4)
integrating factor:
exp((3/t)*dt)=t^3

so y*(t^3)=(cos(t^4))*(t^3)dt)

let t^4=p

4*t^3*dt=dp

so (cos(t^4))*(t^3)dt)=(cos(p)*(1/4)*dp)=sin(p)/4 +c

c=a constant

final solution:-

y*t^3=sin(t^4)*0.25+c

at t=pi^(1/4),y=1

so 1*pi^(3/4)=c

so complete solution:-

y*t^3=sin(t^4)*0.25+pi^(3/4)