Answer to Question #149877 in Differential Equations for Nikhil

Question

Write the equation ydx+ (x-y+x-2y)=0 in the linear form and hence find its solution.

Answer 1

$\displaystyle y\mathrm{d}x+ (x-y+x-2y)\mathrm{d}y =0 \\ y\mathrm{d}x+ (2x - 3y)\mathrm{d}y =0 \\ y\mathrm{d}x+ (2x - 3y)\mathrm{d}y =0 \\ \frac{\mathrm{d}x}{\mathrm{d}y} = \frac{3y - 2x}{y}\\ \frac{\mathrm{d}x}{\mathrm{d}y} = 3 - \frac{2x}{y}\\ \frac{\mathrm{d}x}{\mathrm{d}y} + \frac{2x}{y} = 3\, \textsf{as a linear form}\\ \textsf{Integrating factor}\,\, (IF) = e^{\int \frac{2}{y}\,\mathrm{d}y} = e^{2\ln{y}} = e^{\ln(y^2)} = y^2\\ \frac{\mathrm{d}x}{\mathrm{d}y} + \frac{2x}{y} = 3\\ xIF = \int 3 IF\, \mathrm{d}y\\ xy^2 = \int 3y^2 \, \mathrm{d}y\\ xy^3 = y^3 + C\\ x = 1 + Cy^{-3}\, \textsf{is the solution to the ODE}$