Toronto Math Forum

The problem is given:
$$u_{tt} - u_{xx} = (x^2 -1)e^{-\frac{x^2}{2}}$$
$$u(x,0) = -e^{-\frac{x^2}{2}}, u_t(x,0) = 0$$

I have already got the general solution as followed, but I have trouble solving the integral,
$$\int_{0}^{t} \int_{x-t+s}^{x+t-s} (y^2-1)e^{-\frac{x^2}{2}}dyds$$
and I tried $\Delta$ method, but it seems to make the equation more complex. Professor, could you please give a hint of solving this problem?

Professor, may I ask how to derive all the five conditions mentioned above? Why it's not necessary to consider the condition like ${x+ct>1, x-ct<-1}$ or ${x-ct>1, x+ct,-1}$ something like this?

Given the question $u_t + 3u_x -2u_y = xyu$. My steps are:
$$\frac{dt}{1} = \frac{dx}{3} = \frac{dy}{-2} = \frac{du}{xyu} \Rightarrow x=C_1+3t, y=C_2-2t$$
$$xydt = \frac{du}{u} \Rightarrow (C_1+3t)(C_2-2t)dt = \frac{du}{u}$$
$$\ln u = C_1C_2 t + \frac{t^2}{2} (3C_2 - 2C_1) - 2t^3 +\phi(C_1, C_2)$$
I know then I need to insert base $e$ into both sides to get $u$, but it would be complex. I wonder is there any simple way to solve this? Or do I have something wrong in above equations so that leads to the complex solution?

I guess that since $u$ is of $x$ and $x$ is of $t$, they are not independent and you need to use $\partial$ in your steps. Also, there should be chain rule in this case, so you cannot simply take integrals as in your second equation.

I have trouble solving this problem: $yu_x - xu_y = x^2$. I know the characteristic equation is $\frac{dx}{y} = \frac{dy}{-x} = \frac{du}{x^2}$ and then have $C = \frac{x^2}{2} + \frac{y^2}{2}$. Then the following should be the integration relative to $du$, but either $\frac{du}{dx}$ or $\frac{du}{dy}$ will contain not only one variable, like $\frac{du}{dx} = \frac{x^2}{y}$ contain both $x$ and $y$. I wonder if $x$ and $y$ are independent here. If not, should I rewrite the expression $C = \frac{x^2}{2} + \frac{y^2}{2}$ in order to get the expression of y in terms of x , and then applies it into the integration relative to $du$? Any reply would be appreciated.

I have difficulties solving this problem: $u_x + u_y + u = e^{x+2y}$. I get $\frac{dy}{dx} = 1$ from the characteristic equation and have no ideas what should be followed. Any reply would be appreciated.

I have a guess but also not quite sure if I'm in the right way. Integrate the equation you got
$$\int_ {}{}\frac{(uu_x)_y}{uu_x} dy = \int {}{}\frac{u_y}{u}dy \qquad\qquad\qquad\qquad \color{red}{\text{Error in the left part; the rest incorrect. V.I.}}$$
$$ \ln uu_x = \ln u + f(x)$$
where $f$ is any functions that is only of $x$,
$$uu_x = u \cdot g(x)$$
where $g$ is a function of $f$ mentioned above. Divide $u$ from both sides,
$$ u_x = g(x)$$
then integrate both sides,
$$ u = G(x) + h(y)$$
where $G'(x) = g(x)$, and $h$ is any function that is only of y.

Sorry.. I think the following one is the correct solution. I had wrong arg computation above.
This one maybe the correct answer.