Method of Characteristics – Lagrange-Charpit Equations

Question

I need to solve the following PDE with initial condition $U(x,0)=U_0(x)$. Once this is one of my first times, I'd like to get second opinions.

Many Thanks.

\begin{equation} \partial_t U + (\text{cos}(t)+1)\partial_x U = -2,
\end{equation}

The characteristic curves are given by Lagrange-Charpit equations:
\begin{equation}
\dfrac{dt}{1}=\dfrac{dx}{(\text{cos}(t)+1)}=\dfrac{dU}{-2},\qquad(i)
\end{equation}

Following: \begin{equation} \dfrac{dU}{dt}=-2.\qquad(ii)
\end{equation}

Solving this EDO \begin{equation} U(x(t),t)=-t+c_1\qquad(iii)
\end{equation}

By other hand: \begin{equation} \dfrac{dx}{dt}=(\cos(t)+1).\qquad(iv)
\end{equation}

So: \begin{equation} x(t)= t+\sin (t)+c_2.\qquad(v) \end{equation}

Setting $x(0)=x_0$, we get $c_2=x_0$. So, the characterhistics are
given by: \begin{equation}x(t)=t+\sin (t)+x_0.\qquad(vi)\end{equation}

The whole solutions is given by:
\begin{equation}
U(x,t)=-t+U_0(x_0)=-t+U_0\bigg(x-t+\sin (t)\bigg).\qquad(vii)
\end{equation}

Answer 1

Note that while $U(x,0) = U_0(x)$, the PDE itself isn't satisfied. \begin{align} U_t + (1 + \cos t)U_x &= -1 + U_0'(x - t + \sin t)(-1 + \cos t) + U_0'(x - t + \sin t)(1 + \cos t) \\ &= -1 + 2\cos t \ U_0'(x - t + \sin t) \\ &\not \equiv -2. \end{align}

But $$ \frac{\mathrm{d}U}{\mathrm{d}t} = -2 \implies U(x(t),t) = -2t + c_1 $$ and $$ x(t) = t + \sin t + x_0 \implies x_0 = x(t) - t - \sin t. $$