partial differential equation - 10

[logistic_guy]

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

\(\displaystyle ax^2\frac{\partial w}{\partial x} + by^2\frac{\partial w}{\partial y} = cw^2\)

 

[logistic_guy]

If you wanna live longer, solve partial differential equations.

\(\displaystyle \frac{dx}{ax^2} = \frac{dy}{by^2} = \frac{dw}{cw^2}\)


\(\displaystyle \frac{dx}{ax^2} = \frac{dy}{by^2}\)


\(\displaystyle \int \frac{dx}{ax^2} = \int \frac{dy}{by^2}\)


\(\displaystyle -\frac{1}{ax} = -\frac{1}{by} + C_1\)


\(\displaystyle \frac{1}{by} -\frac{1}{ax} = C_1\)

Then,

\(\displaystyle \Phi(C_1) = \Phi\left(\frac{1}{by} -\frac{1}{ax}\right) = C_2\)

 

[logistic_guy]

\(\displaystyle \frac{dx}{ax^2} = \frac{dy}{by^2} = \frac{dw}{cw^2}\)

We continue.

\(\displaystyle \frac{dx}{ax^2} = \frac{dw}{cw^2}\)


\(\displaystyle \int \frac{dx}{ax^2} = \int\frac{dw}{cw^2}\)


\(\displaystyle -\frac{1}{ax} = -\frac{1}{cw} + C_2\)


\(\displaystyle \frac{1}{cw} = \frac{1}{ax} + C_2\)


\(\displaystyle cw = \left[\frac{1}{ax} + C_2\right]^{-1}\)


\(\displaystyle w = \frac{1}{c}\left[\frac{1}{ax} + C_2\right]^{-1}\)

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

\(\displaystyle w = \frac{1}{c}\left[\frac{1}{ax} + \Phi\left(\frac{1}{by} -\frac{1}{ax}\right)\right]^{-1}\)