How to Solve this Particular Euler Differential Equation? I was asked to solve an Euler ODE:
Here's my work so far:

I'm asked to solve the following Euler DE using the method of variation of parameters:
  $$x^2y''-y=(x^2+1)\sin{x}$$
  I already determined the homogeneous solution which resulted in:
  $$y_h=c_1 x^{\frac{\sqrt{5}+1}{2}}+c_2 x^{\frac{-\sqrt{5}+1}{2}}$$
  I'm having trouble finding the particular solution though...
  $$y_p=u_1 y_1+u_2 y_2$$
  $$u_1=-\int \frac{R(x)\cdot y_2}{W}~dx \qquad \qquad  W=\text{wronskian}=-\sqrt{5}$$
  $$u_1=\frac{1}{\sqrt{5}}\int (x^2+1)\sin{x}\cdot x^{\frac{-\sqrt{5}+1}{2}}~dx$$
  But this integral is too complicated for me to solve. I computed it and found that it gives a gamma function and a really long answer. Is there anything I missed that could simplify this problem?


$$x^2y''-y=(x^2+1)\sin(x)$$
Since $\quad \left(\sin(x)\right)''=-\sin(x)\quad$ obviously $\quad y=-\sin(x)\quad$ is a particular solution. Hence :
$$y(x)=c_1 x^{\frac{1+\sqrt{5}}{2}}+c_2 x^{\frac{1-\sqrt{5}}{2}}-\sin(x)$$
