### Author Topic: Bonus problem 1  (Read 197 times)

#### Victor Ivrii

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##### Bonus problem 1
« on: September 22, 2016, 07:33:09 AM »
Post solution (without screenshots)

Find the general solution $u=u(x,y,t)$

u_t+2 tu_x + xu_y=0

and also a solution satisfying initial condition $u|_{t=0}=\sin (x+y)$.

#### XinYu Zheng

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##### Re: Bonus problem 1
« Reply #1 on: September 22, 2016, 09:03:50 AM »
First, we find the characteristics along which $u$ is constant:

dt=\frac{dx}{2t}=\frac{dy}{x}

Combining the first and second, we have $t^2=x+C_1$. So $x=t^2-C_1$. Subbing this in the last fraction and then combining with the first we obtain $t^3/3-C_1t=y+C_2$, or subbing in for $C_1$, $-2t^3/3+xt-y=C_2$. So our curve in $\mathbb{R}^3$ is described by the intersection of two surfaces, namely $t^2=x+C_1$ and $-2t^3/3+xt-y=C_2$. Since $u$ is constant along this curve, the general solution takes on the form of $u(x,y,t)=\phi(C_1,C_2)=\phi(t^2-x,-2t^3/3+xt-y)$. Now using the initial condition, we have $u|_{t=0}=\sin(x+y)=\phi(-x,-y)$. Thus we have $\phi(x,y)=\sin(-x-y)$. So we have the solution:

u(x,y,t)=\sin(x-t^2+2t^3/3-xt+y)

A quick check shows that this is indeed a solution to the ODE and satisfies the initial conditions.
« Last Edit: September 22, 2016, 12:58:47 PM by XinYu Zheng »