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.