Quick question, I cannot seem to figure to out where I am making a mistake.
The question is: $$ u_x + sin(x) u_y = u $$ with boundary conditions: $$u (0,y) = y^2$$
So my approach was to do:
$$ \frac{dy}{dx} = sinx, \frac{du}{dx} = u$$
Solving for the homogenous equation is: $$y + cos(x) = y_0$$ $$ u(x,y) = f(y + cos(x))$$
Also I know that solving the du/dx equation: $$ u(x,y(x)) = ke^x$$ $$ k = u(0,y_0) = (y_0)^2$$ Plugging in the $y_0$ value: $$k = (y + cos(x))^2$$ Then the solution should be: $$ (y+cos(x))^2 e^x$$
But the solution says it should be: $$ (y + cos(x) -1)^2 e^x$$, not sure where I went wrong.
Thank you for any guidance.
The book is right. The reason is that, on a point $(0,y_0)$ on the initial curve, the combination $y+\cos x$ is equal to $y_0+1$ and then on the characteristic $u=u(y_0)=u(y+\cos x-1)$, etc.