Question

Solve the ODE by Linear ODE method:

x dy/dx + 3y = 6x

Answer 1

x $\frac{dy}{dx}$ + 3y = 6x

Expressing the ODE in the form of linear differential equation, $\frac{dy}{dx} + py= q$ , where p, q are constants or function of x only we get

$\frac{dy}{dx} + \frac{3}{x}y= 6$

Integrating factor is $e^{\int {\frac{3}{x}dx}}= e^{3ln{x}}=e^{ln{x³}}= x³$

Multiplying both sides by integrating factor , x³

$x³(\frac{dy}{dx} + \frac{3}{x}y)= 6x³$

=> $x³\frac{dy}{dx} + {3x²}y)= 6x³$

=> $x³dy+ {3x²}ydx= 6x³dx$

=> $d{(x³y)} = 6x³dx$

Integrating both sides

$\int d{(x³y)} = 6\int x³dx+C'$ , where $C'$ is integration constant.

=> $x³y = 6. \frac{x⁴}{4} + C'$

=> $x³ y = \frac{3x⁴}{2}+ C'$

=> $2x³y-3x⁴ = 2C'$

=> $2x³y-3x⁴ = C$ where $C = 2C'$

This is the solution of the given differential equation.