I have to solve the following pde using Laplace transforms:
$xw_x + w_t= xt$ i.c: w(x,0)= 0
Firstly, transforming the above wrt t, i get: $\bar{w_x} + s\bar{w}/x = 1/s^2$
But, in the textbook, the transformation is given as : $\bar{w_x} + s/x = 1/s^2$ Why is there no $\bar{w}$ on the LHS in the textbook answer?
definition of Laplace transform from textbook: $\int_{0}^{\infty} e^{-st}u(t)dt$
Lalpace transform used: $L(u')= s\bar{u} - u_0$