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APM346-2021S => APM346--Tests & Quizzes => Quiz 1 => Topic started by: Jiaqi Bi on January 28, 2021, 12:28:16 PM

Title: Quiz 1-0201 C Problem 2
Post by: Jiaqi Bi on January 28, 2021, 12:28:16 PM
Problem 2: Find the general solutions to the following equation:
$$u_{xyz}=x+y+z$$
Solution:

\begin{align*}
u_{xyz}&=x+y+z\\
u_{xy}&=xz+yz+\frac{1}{2}z^2+f(x,y)\\
u_{x}&=xyz+\frac{1}{2}y^2z+\frac{1}{2}yz^2+f'(x,y)+g(x,z)\\
u&=\frac{1}{2}x^2yz+\frac{1}{2}xy^2z+\frac{1}{2}xyz^2+f''(x,y)+g'(x,z)+h(y,z)\\
&=\frac{1}{2}x^2yz+\frac{1}{2}xy^2z+\frac{1}{2}xyz^2+\phi (x,y)+G(x,z)+h(y,z)\ \ \ \ \ \ \ \  (\text{Say}\ \phi (x,y)=f''(x,y), G(x,z)=g'(x,z))
\end{align*}