We introduce the indepenedent variable \(\displaystyle s\) such that:
\(\displaystyle \frac{dx}{ds} = a\)
\(\displaystyle \frac{dy}{ds} = b\)
\(\displaystyle \frac{dw}{ds} = cw + (\beta x^n + \lambda y^m)w^k\)
Then,
\(\displaystyle dx = a \ ds\)
\(\displaystyle dy = b \ ds\)
Or
\(\displaystyle \int dx = \int a \ ds\)
\(\displaystyle \int dy = \int b \ ds\)
Or
\(\displaystyle x = as + x_0\)
\(\displaystyle y = bs + y_0\)
Then,
\(\displaystyle \frac{dw}{ds} = cw + (\beta (as + x_0)^n + \lambda (bs + y_0)^m)w^k\)
Let \(\displaystyle v = w^{1-k}\).
Then,
\(\displaystyle \frac{dv}{ds} = \frac{dw^{1-k}}{ds} = \frac{dw^{1-k}}{dw}\frac{dw}{ds} = (1- k)w^{-k}[cw + (\beta (as + x_0)^n + \lambda (bs + y_0)^m)w^k]\)
Simplify.
\(\displaystyle \frac{dv}{ds} = c(1- k)w^{1-k} + (1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)\)
Or
\(\displaystyle \frac{dv}{ds} - c(1- k)v = (1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)\)
This is just an ordinary differential equation and it can be solved by integrating factor.
Let \(\displaystyle F(s)\) be the integrating factor, then
\(\displaystyle F(s) = \text{exp}\bigg(\int c(1 - k) \ ds\bigg) = \text{exp}\bigg(c(1 - k)s \bigg)\)
Divide it by each term in the differential equation.
\(\displaystyle \frac{1}{F(s)}\frac{dv}{ds} - \frac{c(1- k)v}{F(s)} = \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)}\)
Or
\(\displaystyle \frac{d}{ds}\left(\frac{v}{F(s)}\right) = \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)}\)
Or
\(\displaystyle \int\frac{d}{ds}\left(\frac{v}{F(s)}\right) \ ds = \int \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)} \ ds\)
Or
\(\displaystyle \frac{v}{F(s)} + D = \int \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)} \ ds\)
Or
\(\displaystyle v = -DF(s)+ F(s)\int \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)} \ ds\)
Or
\(\displaystyle v = C_2F(s)+ F(s)\int \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)} \ ds\)
Or
\(\displaystyle v = \Phi(bx - ay)F(s) + F(s)\int \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)} \ ds\)
We know \(\displaystyle v = w^{1-k}\)
Then, the general solution to the partial differential equation is:
\(\displaystyle w = v^{\frac{1}{1 - k}} = \left(\Phi(bx - ay)F(s) + F(s)\int \frac{(1- k)(\beta (as + x_0)^n + \lambda (bs + y_0)^m)}{F(s)} \ ds\right)^{\frac{1}{1-k}}\)
where
\(\displaystyle F(s) = \text{exp}\bigg(c(1 - k)s \bigg)\)
\(\displaystyle x = as + x_0\)
\(\displaystyle y = bs + y_0\)
