Find the solution of the given initial value problem Ty 8y

Find the solution of the given initial value problem. Ty\' + 8y = t^2 - t + 1, y(1) = 1/8, t > 0 y(t) = t^2/4 - t/3 + 1/2 - 7/24t^8

Solution

we can rewrite as

y\'+8y/t=t-1+1/t

Integrating factor is t^8

Multiplying by this integrating factor gives

t^8y\'+8yt^7=t^9-t^8+t^7

(yt^8)\'=t^9-t^8+t^7

Integrating gives

yt^8=t^10/10-t^9/9+t^8/8+C

y(1)=1/8

1/8=1/10-1/9+1/8+C

C=1/9-1/10=1/90

yt^8=t^10/10-t^9/9+t^8/8+1/90

y=t^2/10-t/9+1/8+1/(90t^8)