Solve this integrodifferential equation y integral 0 yvt v

Solve this integro-differential equation: y\' - integral _0 y(v)(t - v)dv = 1 + t with y(0) = 1

Solution

Let F(s) denote the Laplace transform of y(t).

Now L[y\'(t)]= sF(s)-y(0)= sF(s)-1

The integral is the convolution of y(t) and the function g(t) =t.

L (y(t)t) = F(s)/s²

Taking Laplace transform throughout , we get

sF(s)-1-F(s)/s² = 1/s+1/s²

Simplifying this , F(s) =1/s-1

so f(t) =e^t