Can You Help Me Solve This Non-Elementary Differential Equation?

[bishy]

Homework Statement

2y\prime\prime +2y\prime + y = 4 \sqrt{x}

The Attempt at a Solution

charecteristic equation: x^2+x+\frac{1}{2}
roots: \frac{1}{2}\pm\frac{1}{2}i

homogenous solution: a \sin{\frac{1}{2}x} + b \cos{\frac{1}{2}x}

Wronskian: \left(\begin{array}{cc}\sin{\frac{1}{2}x}&\cos{\frac{1}{2}x}\\\frac{1}{2}\cos{\frac{1}{2}x}&-\frac{1}{2}\sin{\frac{1}{2}x}\end{array}\right) = -\frac{1}{2}

It would be nice to know if up to here, everyone else gets the same answer. After this I get into non elementary functions which is no where near the level of difficulty included within the course I'm taking. I haven't attempted to solve what comes next, frankly because I have no clue where to even begin. I think I probably made a mistake above, if someone can point me in the right direction that would be awesome. The method used is variation of parameters.

y_{1}=\sin{\frac{1}{2}x}

y_{2}=\cos{\frac{1}{2}x}

u_{1} = \int{4\sqrt{x}\cos{\frac{1}{2}x}dx}

u_{2} = \int{-4\sqrt{x}\sin{\frac{1}{2}x}dx}

ick

 

[jeffreydk]

With x^2+x+\frac{1}{2} as your characteristic equation, I think your roots should be -\frac{1}{2}\pm\frac{1}{2}i. And then with the complex roots, the homogeneous solution should be in the form of

y=c_1e^{\lambda t}\cos(\mu t)+c_2e^{\lambda t}\sin(\mu t)

where the roots come from the form of \lambda \pm i\mu. Your form was missing the exponential term. Hope that helps a bit.

 

[Defennder]

Yeah, jeffreydk is right. Your roots are incorrect and you neglected the exponential.