# differential equations – Solving a PDE with boundary and initial conditions

today I was studying partial differential equations and I tried to check if my solution was correct, I thought Mathematica code would be easy, but it wasn’t.
Here is my problem:

Solve $$u_{xx}-2tu-u_t=0$$ for $$0 and $$t>0$$ with the boundary conditions $$u_x(0,t)=u(1/2,t)=0$$ and initial conditions $$u(x,0)=1-2x$$.

By hand I obtained
$$u(x,t)=frac{8}{pi^2}sum_{n=0}^{infty}frac{1}{(2n+1)^2}e^{-t^2-(2n+1)^2pi^2t}cos((2n+1)pi x)$$

Then I tried with the following code in Mathematica:

``````pde = D(u(x, t), {x, 2}) - 2 t*u(x, t) == D(u(x, t), {t, 1});
ic = {u(x, 0) == 1 - 2 x};
bc = {u(1/2, t) == 0, Derivative(1, 0)(u)(0, t) == 0};
sol = First@(u(x, t) /. DSolve({pde, ic, bc}, u(x, t), {x, t}));
``````

I don’t know why I am getting error. Thanks in advance for your help.