partial differential equation - 2

[logistic_guy]

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

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

 

[logistic_guy]

Let \(\displaystyle v = ax + by + cw\)

\(\displaystyle \frac{\partial v}{\partial x} = a + c\frac{\partial w}{\partial x}\)


\(\displaystyle \frac{\partial v}{\partial y} = b + c\frac{\partial w}{\partial y}\)

Substitute this in the original differential equation.

\(\displaystyle \frac{1}{c}\left(\frac{\partial v}{\partial x} - a\right) + \frac{k}{c}\left(\frac{\partial v}{\partial y} - b\right) = v^n + s\)

Simplify.

\(\displaystyle \left(\frac{\partial v}{\partial x} - a\right) + k\left(\frac{\partial v}{\partial y} - b\right) = cv^n + cs\)


\(\displaystyle \frac{\partial v}{\partial x} - a + k\frac{\partial v}{\partial y} - kb = cv^n + cs\)


\(\displaystyle \frac{\partial v}{\partial x} + k\frac{\partial v}{\partial y} = a + bk + cv^n + cs\)

This differential equation looks familiar.

 

[logistic_guy]

\(\displaystyle \frac{\partial v}{\partial x} + k\frac{\partial v}{\partial y} = a + bk + cv^n + cs\)

This differential equation looks familiar.

\(\displaystyle \frac{dx}{1} = \frac{dy}{k} = \frac{dv}{a + bk + cv^n + cs}\)


\(\displaystyle \int k \ dx = \int \ dy\)


\(\displaystyle kx + C_1 = y\)


\(\displaystyle y - kx = C_1\)

Then,

\(\displaystyle \Phi(C_1) = \Phi(y - kx) = C_2\)

 

[logistic_guy]

\(\displaystyle \frac{dx}{1} = \frac{dy}{k} = \frac{dv}{a + bk + cv^n + cs}\)

\(\displaystyle \frac{dx}{1} = \frac{dv}{a + bk + cv^n + cs}\)


\(\displaystyle \int\frac{dx}{1} = \int\frac{dv}{a + bk + cv^n + cs}\)


\(\displaystyle x + C_2 = \int\frac{dv}{a + bk + cv^n + cs}\)


\(\displaystyle x + \Phi(y - kx) = \int\frac{dv}{a + bk + cv^n + cs}\)

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

\(\displaystyle \int\frac{dv}{a + bk + cv^n + cs} = x + \Phi(y - kx)\)

where \(\displaystyle v = ax + by + cw\)

Or

\(\displaystyle w(x,y) = \frac{v - ax - by}{c}\)