Problem 3

[Djirar]

Hello,

For this question I managed to find a solution that satisfies the conditions, but that only depends on X and a constant.

In general, if I find a solution that satisfies all conditions is this solution correct regardless of the method used to find it?

[Victor Ivrii]

Hello,

For this question I managed to find a solution that satisfies the conditions, but that only depends on X and a constant.

In general, if I find a solution that satisfies all conditions is this solution correct regardless of the method used to find it?

You need to find all solutions and justify that there are no other solutions. The method of characteristics we studied ensures this if correctly applied. Your home-brewed method may not.

[Djirar]

I did use the method of characteristics, but I parametrized x and y in terms of s and integrated with respect to s . The thing is my solution doesn't depend on Y, is this ok ?

[Victor Ivrii]

I did use the method of characteristics, but I parametrized x and y in terms of s and integrated with respect to s . The thing is my solution doesn't depend on Y, is this ok ?

I have no idea what is $s$. TA who will check your paper probably has no idea either. You need to return to the original coordinates $x,y$.

[Djirar]

I got it, thank you for your help.

[Aida Razi]

Solution is attached!

[James McVittie]

Find the solution:

$u_{x}+3u_{y}=xy$

$u_{x=0}=0$

By examining Integral Lines:

$\frac{1}{dx}=\frac{3}{dy}=\frac{xy}{du}$ <very ugly form. Never use it $\color{blue} {\frac{dx}{1}=\frac{dy}{3}=\frac{du}{xy}}$

Then from the first equality:

$3x=y+C$ where C is some constant.

$y=3x-C$

Then again from the Integral Lines:

$dx(xy)=du$

$dx(x(3x-C))=du$

$u=x^{3}-\frac{C}{2}x^{2}+C_{1}$

Then by the initial condition:

$u(x=0)=0$

$C_{1}=0$

Therefore,$C=3x-y$

Therefore,

$u(x,y)=x^{3}-\frac{1}{2}x^{2}C$

$u(x,y)=x^{3}-\frac{1}{2}x^{2}(3x-y)$

$u(x,y)=-\frac{1}{2}x^{3}+\frac{1}{2}x^{2}y$

Check:

$u_{x}(x,y)=-\frac{3}{2}x^{2}+xy$

$u_{y}(x,y)=\frac{1}{2}x^{2}$

$u_{x}+3u_{y}=xy$

[Victor Ivrii]

Now it is correct.