Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions and one particular solution of the homogeneous equation:

Question

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions of this equation:

$y_2=\ln(x)$

$y_1=x+\ln(x)$

and one particular solution of the homogeneous equation:

$y_3=e^{2x}$

I think we can take two linear independent solutions: $x$ and $\ln(x)$ make $Y=c_1x+c_2\ln(x)+e^{2x}$

but i don't know what to do next

Answer 1

if $y_1 = \ln x$ and $y_2 = x + \ln x$ are particular solution, then the difference $y_2 - y_1 = x$ is a solution to the associated homogeneous equation. now we have two linearly independent solutions $y = x, y = e^{2x}.$ they satisfy $$\frac{d^2y}{dx^2} = 0 \text{ and } \frac{dy}{dx} - 2y = 0$$ now we can form a third order differential equation $$0 = \frac{d^2}{dx^2}(\frac{d}{dx} - 2)y = \frac{d^3y}{dx^3} - 2\frac{d^2y}{dx^2}$$

now we will get the forced equation and the force so that $\ln x$ is a particular solution.

$(\frac{d^3}{dx^3} -2\frac{d^2}{dx^2}) \ln x = {2 \over x^3} + {2 \over x^2} = {2(x+1) \over x^3}.$ here is the equation satisfying all the requirements:

$$\frac{d^3y}{dx^3} - 2\frac{d^2y}{dx^2} = {2(x+1) \over x^3}.$$