I need to solve this equation
I know how to solve such equations of the form, $u_t=u_{xx}-au$ using the transformation $$u=e^{-at}w(x,t).$$
But the problem is the the coefficient is in terms of $t$, it is not constant, so i think this transformation wont work.
Can you suggest any hint. I can solve it on my own.
Can this be solved without having any BCs , coz only IC is given.
$$(t+1)u_t-u=(t+1)u_{xx}$$ $$\dfrac {(t+1)u_t-u}{(t+1)^2}=\dfrac {u_{xx}}{t+1}$$ $$\left(\dfrac u {t+1} \right)_t=\left (\dfrac {u}{t+1}\right)_{xx}$$ $$w_t(x,t)=w_{xx}(x,t)$$ Where $w(x,t)=\dfrac {u(x,t)} {t+1}$