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APM346--2019 => APM346--Lectures & Home Assignments => Home Assignment 1 => Topic started by: Zhiman Tang on January 18, 2019, 07:01:19 PM

Title: problem 4 (25)
Post by: Zhiman Tang on January 18, 2019, 07:01:19 PM

u_xy = u_xu_y
I use the hint and divide both sides by u_x. I get u_x = e^u * f(x). I am stuck there. Could anybody help me out?

Title: Re: problem 4 (25)
Post by: Wanying Zhang on January 18, 2019, 08:04:40 PM

I would start from the beginning.
$$\frac{u_{xy}}{u_x} = \frac{u_x u_y}{u_x} \Rightarrow \frac{u_{xy}}{u_x} = u_y$$
Integrate both sides,
$$\ln{u_x} = u + f(x)$$
$$u_x = e^{u+f(x)} = e^u \cdot g(x)$$
$$\frac{\partial u}{\partial x} = g(x)e^u$$
$$\frac{\partial u}{e^u} = g(x)\partial x$$
Integrate both sides,
$$-e^{-u} = G(x) + h(y)$$
$$u(x,y) = -\ln (-G(x) - h(y))$$
OK. V.I.