Find the Characteristic Curves and general solution of a PDE

Question

Find the Characteristic Curves and general solution of

$\frac{∂u}{∂x} + 2\frac{∂u}{∂y} = 0.$

Also find the particular solution when

$u(x, 0) = e^x$.

So far I have that the characteristic curve is $y=2x+C$ and the solution of the PDE is $u=f(2x-y)$. (Using the general formula $f(bx-ay)=0$)

And that

$f(2x)=e^x$

If we let $s=2x$ then $e^\frac{s}{2}=f(s)$

Hence $u(x,y)=f(2x-y)=\sqrt{e}(2x-y)$ is the Particular solution.

Im unsure if my Particular Solution is right?

Answer 1

You calculus is right up to $u=f(2x-y)$ and $f(2x)=e^x$. There is a mishmash after.

Change of variable : $\quad X=2x \quad\to\quad f(X)=e^{\frac{X}{2}}$

Now, the function $f$ is known. Put $(2x-y)$ into it : $$u=e^{\frac{2x-y}{2}}$$