Solve the differential equation for y(t) in terms of x(t)

Question

Consider the following differential equation: $$ (t+1)y'(t) + y(t) = x'(t) + x(t) \quad (t > 0) $$ With initial conditions $$ y(0) = 0 \quad x(0) = 0 $$

Solve for $y(t)$ in terms of $x(t)$.

Also, (although I can probably figure out this part on my own) - is the system linear? Causal?

Thanks

Answer 1

Define $\tilde y(t)=(t+1) y(t)$. Then $$ \tilde y^\prime =x^\prime +x $$ Then $$ \tilde y(t)=x(t)+\int_0^t x(t) dt+C $$ and $$ y(t)=(x(t)+\int_0^t x(t) dt+C)/(t+1) $$ From initial condition $C=0$.