Answer to Question #233241 in Differential Equations for sam

Question

Solve the equation (dy/dx) − y = e^xy²

Answer 1

$\frac{dy}{dx} - y = (e^x)y^2$ ...(1)

taking $y = \frac{1}{v}$

$\frac{dy}{dx}=-\frac{1}{v^2}.\frac{dv}{dx}$

From (1)

$-\frac{1}{v^2}.\frac{dv}{dx}-\frac{1}{v}=e^x.\frac{1}{v^2}$

$\frac{dv}{dx}+v=-e^x...(2)$

The integrating factor $=e^{\int1dx}=e^x$

Thus, the solution of (2) is

$v.e^x=\int(-e^x).e^xdx+c$

$v.e^x=-\int e^{2x}dx+c$

$v.e^x=-\frac{ e^{2x}}{2}+c$

$\frac{e^x}{y}=-\frac{ e^{2x}}{2}+c$

$\frac{1}{y}=\frac{ -e^{x}+2c.e^{-x}}{2}$

$y=\frac{2}{ -e^{x}+2c.e^{-x}}$