Toronto Math Forum
APM3462021S => APM346Tests & Quizzes => Quiz 1 => Topic started by: Gavrilo Milanov Dzombeta on January 29, 2021, 10:55:56 AM

$$\text{Find the general solutions to the following equation: }
u_{xxy}=x\cos(y)$$
\begin{gathered}
\nonumber
u_{xxy}=x\cos(y)\\
u_{xx}=x\sin(y) + f(x)\\
u_{x}=\frac{x^2}2\sin(y) + \tilde{f}(x) + h(y) ;\tilde{f}_x=f \\
u= \frac{x^3}6\sin(y) + \tilde{\tilde{f}}(x) + xh(y) + g(y);\tilde{\tilde{f}}_{xx}=f \\
\therefore u(x,y)=\frac{x^3}6\sin(y) + \tilde{\tilde{f}}(x) + xh(y) + g(y).\\
\text{Where $f(x),h(y),g(y)$ are arbitrary functions, and $\tilde{\tilde{f}}$ is twice differentiable.}
\end{gathered}

Try to avoid highriser notations. Several years ago I was a referee for a paper which used notations like this $\widehat{\dot{\widetilde{\mathcal{D}}}}$ and sometimes this little pesky dot was missing :D