Toronto Math Forum
APM3462019 => APM346Lectures & Home Assignments => Home Assignment 2 => Topic started by: Wanying Zhang on January 20, 2019, 12:53:01 AM

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.

Sure, $x$ and $y$ are not independent along integral curves. To proceed you need to parametrize the integral curve. Think: what is the best way to parametrize it?

I tried to use the trigonometric identities to solve this problem and think I got the correct answer, which is $\frac{1}{2} xy + (\frac{1}{2})(x^2 + y^2)arcsin(\frac{y}{\sqrt{x^2 + y^2}})$. But then I'm confused about "In one instance solution does not exist" at end of this problem because I obtain solutions for all 4 subproblems, except two of them with arbitrary function. I wonder which subproblem may not have solutions.

I obtain solutions for all 4 subproblems, except two of them with arbitrary function.
No, your solution is incorrect because it is not a continuous singlevalued function. I gave you a hint: what is the natural parameter along integral curves?

Heller professor
By parametrizing Y in terms of X, do we need to put a plus/minus sign in front of the root? If yes, does that mean when we put down the final solution, we need to include plus and minus as well?

There is NO root. You need to parametrize before integration

There is NO root. You need to parametrize before integration
but, when we parametrize Y in terms of X, don't we have to use $$y^2+x^2=C$$
and thus $$y= +/ \sqrt{Cx^2}$$?

By parameterizing, the professor means to express x and y in terms of some other parameter. Since the integral curves are circles, think about what is normally used to parameterize a circle in a very easy way  it's essentially a change of variable.