1. ## help with y''+ycoshx=0

i have twisted my mind but don't know!! It's not x=e^t, or y=e^t, and can't think of an appropriate variation of the type y=v(x)f(x).

what to do?? any hints?
pepsigirl

2. Whoa. WolframAlpha gave this intimidating answer. The Mathieu cosine and sine functions appear to be the solutions to this DE. Not very pretty, I'm afraid. And I don't know much more than what I've just written. What class is this for, anyway?

3. The DE is linear and 'incomplete' and its general solution is...

$\displaystyle y(x)= c_{1}\ \varphi_{e} (x) + c_2 \ \varphi_{o} (x)$ (1)

... where $\displaystyle \varphi_{e} (x)$ and $\displaystyle \varphi_{o} (x)$ are respectively an 'even' and an 'odd' function that we suppose analytic so that is...

$\displaystyle \displaystyle \varphi_{e} (x) = \sum_{n=0}^{\infty} a_{n}\ x^{2n}$

$\displaystyle \displaystyle \varphi_{o} (x) = x\ \sum_{n=0}^{\infty} b_{n}\ x^{2n}$ (2)

Let's consider first $\displaystyle \varphi_{e} (x)$, for which we can hypothesize that is $\displaystyle a_{0}=1$. Is...

$\displaystyle \displaystyle \varphi^{''}_{e} (x) = \sum_{n=1}^{\infty} 2n\ (2n-1)\ a_{n}\ x^{2n-2}$ (3)

... and remembering that is...

$\displaystyle \displaystyle \cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$ (4)

... we arrive to write...

$\displaystyle \displaystyle \varphi_{e} (x)\ \cosh x= \sum_{n=0}^{\infty} x^{2n}\ \sum_{k=0}^{n} \frac{a_{k}}{\{2\ (n-k)\}!}$ (5)

Combining (3), (5) and the DE we are able to find the $\displaystyle a_{n}$...

$\displaystyle a_{0}=1$

$\displaystyle 2\ a_{1} + a_{0} = 0 \rightarrow a_{1} = - \frac{1}{2}$

$\displaystyle \displaystyle 12\ a_{2} + \frac{a_{0}}{2} + a_{1}=0 \rightarrow a_{2} = 0$

$\displaystyle \displaystyle 30\ a_{3} + \frac{a_{0}}{6!} + \frac{a_{1}}{4!} + \frac{a_{2}}{2} + a_{3} = 0 \rightarrow a_{3}= \frac{1}{1860}$

... so that is...

$\displaystyle \displaystyle \varphi_{e} (x)= 1 - \frac{x^{2}}{2} + \frac{x^{6}}{1860} + ...$ (6)

To be continued in succesive post...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. Regarding the 'odd function' $\displaystyle \varphi_{o} (x)$ we proceed as the $\displaystyle \varphi_{e} (x)$ and obtain...

$\displaystyle \displaystyle \varphi^{''}_{o} (x) = \sum_{n=1}^{\infty} 2n\ (2n+1)\ b_{n}\ x^{2n-1}$ (1)

... and...

$\displaystyle \displaystyle \varphi_{o} (x)\ \cosh x = \sum_{n=0}^{\infty} x^{2n+1}\ \sum_{k=0}^{n} \frac{b_{k}}{\{2\ (n-k)\}!}$ (2)

Now from (1), (2) and the DE we obtain the $\displaystyle b_{n}$ as follows...

$\displaystyle b_{0}=1$

$\displaystyle \displaystyle 6\ b_{1} + \frac{b_{0}}{2} + b_{1} = 0 \rightarrow b_{1} = -\frac{1}{14}$

$\displaystyle \displaystyle 20\ b_{2} + \frac{b_{0}}{4!} + \frac{b_{1}}{2} + b_{2} =0 \rightarrow b_{2}= -\frac{1}{3528}$

... so that is...

$\displaystyle \displaystyle \varphi_{o} (x) = x - \frac{x^{3}}{14} - \frac{x^{5}}{3528} + ...$ (3)

The conclusion is that the DE...

$\displaystyle \displaystyle y^{''} + y\ \cosh x =0$ (4)

... has general solution...

$\displaystyle \displaystyle y(x) = c_{1}\ \varphi_{e} (x) + c_{2}\ \varphi_{o} (x)$ (5)

... where $\displaystyle \varphi_{e} (x)$ and $\displaystyle \varphi_{o} (x)$ have been found in the previous and in the actual post...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$