Find all the solutions

y''(x) + y(x) = cos(x) + xy'(0) + y(0)

I found $\displaystyle \frac{1}{2}$x sen(x) as 1 solution.

I dont know if its the only one.

Printable View

- Aug 9th 2008, 04:04 PMkezmanDifferential equation
Find all the solutions

y''(x) + y(x) = cos(x) + xy'(0) + y(0)

I found $\displaystyle \frac{1}{2}$x sen(x) as 1 solution.

I dont know if its the only one. - Aug 9th 2008, 06:26 PMmr fantastic
You mean you have found $\displaystyle y = \frac{x}{2} \, {\color{red}\sin} x $ as a particular solution to $\displaystyle {\color{red}y''(x) + y(x) = \cos x}$.

You're expected to know that the solution to $\displaystyle y''(x) + y(x) = \cos x + xy'(0) + y(0)$ is the homogeneous solution plus a 'total' particular solution.

The homogeneous solution is the solution to $\displaystyle y''(x) + y(x) = 0$ and you're expected to be able to solve this.

You have a particular corresponding to the $\displaystyle \cos x$ term on the right hand side of the DE. Now you need a particular solution corresponding to the term $\displaystyle y'(0) x + y(0)$. I suggest trying one of the form $\displaystyle y = ax + b$ and getting a and b in terms of y'(0) and y(0). Then the 'total' particular solution is $\displaystyle y = \frac{x}{2} \sin x + ax + b$.

If you're stuck on finding the homogeneous solution or the 'total' particular solution, post what you've done and where you get stuck. - Aug 10th 2008, 12:43 AMwingless
I think it's "sen" instead of "sin" in Spanish (or maybe some other languages).

(I'm telling this because you highlighted that sin, mr fantastic ;p)