# Second order linea non-homogenous differential equation

Given the following equation

$(1)\,\,\,\,x(x-1)y''+xy'-y=0$

Known that $y_{1}=x$

is a solution to his equation (one of many possible)

Now I need to solve the following equation

$(2)\,\,\,\,\,\,\,\,x(x-1)y''+xy'-y=x(x-1)^{2}$

This is a second order non homogenous differential equation with “non constant” coefficients, I would use a reduction of order for the “complementary homogenous” get the homogenous solution solution

$y_{h}=c_{1}(f_{1}(x))+c_{2}(f_{2}(x))$

and then use variation of variables to solver this one.

However is there asimpler way to do this ? can I reduce the order for the whole non-homogenous equation and get this solved ?

If yes(if I can reduce the order straight away, can you please explain when am I allowed to use it )

Thanks, :)

• No, $y_1=x$ isn't solution of $x(x-1)y''+xy'+y=0$ – JJacquelin Aug 6 '16 at 15:34
• @JJacquelin hey, corrected – Pavel Penshin Aug 6 '16 at 16:02

## 1 Answer

Since a particular solution $y_1(x)=x$ is known, the reduction of order is obtained with the change of function :

$$y(x)=xu(x) \quad\to\quad x^2(x-1)u''+x(3x-2)u'=x(x-1)^2$$ Let $v(x)=u'(x)$ $$x(x-1)v'+(3x-2)v=(x-1)^2$$ This is a first order linear non-homogeneous ODE.

I suppose that you can take it from here : First, solve the associated homogeneous ODE $x(x-1)v'+(3x-2)v=0$ which is of separable kind.

Second, solve the non-homogeneous ODE (method of variation of parameter).