partial differential equation - 7

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{Solve.}}\)

\(\displaystyle ax\frac{\partial w}{\partial x} + by\frac{\partial w}{\partial y} = cw^k\)

 

[logistic_guy]

I think at this point we are pros.

\(\displaystyle \frac{dx}{ax} = \frac{dy}{by} = \frac{dw}{cw^k}\)


\(\displaystyle \frac{dx}{ax} = \frac{dy}{by}\)


\(\displaystyle b\int\frac{dx}{x} = a\int\frac{dy}{y}\)


\(\displaystyle b\ln|x| = a\ln|y| + C\)


\(\displaystyle b\ln|x| - a\ln|y| = C\)


\(\displaystyle \ln|x|^b - \ln|y|^a = C\)


\(\displaystyle \ln\frac{|x|^b}{|y|^a} = C\)


\(\displaystyle \frac{|x|^b}{|y|^a} = e^C = C_1\)


\(\displaystyle |x|^b|y|^{-a} = C_1\)

Then,

\(\displaystyle \Phi(C_1) = \Phi(|x|^b|y|^{-a}) = C_2\)

 

[logistic_guy]

\(\displaystyle \frac{dx}{ax} = \frac{dy}{by} = \frac{dw}{cw^k}\)

\(\displaystyle \frac{dx}{ax} = \frac{dw}{cw^k}\)


\(\displaystyle \frac{c}{a}\int\frac{dx}{x} = \int\frac{dw}{w^k}\)


\(\displaystyle \frac{c}{a}\ln|x| + C_2 = \frac{w^{1 - k}}{1 - k}\)


\(\displaystyle \frac{c}{a}\ln|x| + \Phi(|x|^b|y|^{-a}) = \frac{w^{1 - k}}{1 - k}\)

This solution for \(\displaystyle k \neq 1\). For \(\displaystyle k = 1\), the solution is:

\(\displaystyle \frac{c}{a}\ln|x| + \Phi(|x|^b|y|^{-a}) = \ln|w|\)

Then, the general solution to the partial differential equation is:

\(\displaystyle \frac{c}{a}\ln|x| + \Phi(|x|^b|y|^{-a}) = \begin{cases}\frac{w^{1 - k}}{1 - k} & \text{if} \ k \neq 1\\\ln|w| & \text{if} \ k = 1\end{cases} \)