# Second Order DE

## Homework Statement

Not really a homework problem, but more of a project I am working on.

Solve the following differential equation (if possible):

$$C=\frac{\phi ''}{\phi (\phi ' ^2 +1)^2}$$

Where C is a constant, and $\phi$ is a function.

See above.

## The Attempt at a Solution

First, note that
$$\left ( \frac{1}{1+\phi ' ^2} \right ) '=\frac{-2\phi ' \phi ''}{(1+\phi ' )^2}$$

So the original equation can be rewritten:

$$C=\frac{\phi ''}{\phi (\phi ' ^2 +1)^2}=\frac{1}{-2\phi \phi '} \left ( \frac{-2\phi ' \phi ''}{(1+\phi ')^2} \right )=\frac{1}{-2 \phi \phi '} \left ( \frac{1}{1+\phi ' ^2} \right ) '$$

So

$$-2C\phi \phi '=\left ( \frac{1}{1+\phi ' ^2} \right ) ' [/itex] Integrating both sides with respect to the independent variable (call it v) [tex] \int -2C\phi \phi ' \, dv=\int \left ( \frac{1}{1+\phi ' ^2} \right ) ' dv$$

By the fundamental theorem of calculus, we have
$$-C\phi ^2+C_2=\left ( \frac{1}{1+\phi ' ^2} \right ) + C_3$$
$$-C\phi ^2=\left ( \frac{1}{1+\phi ' ^2} \right ) + C_4$$

Rewriting:
$$\phi ' = \sqrt{\frac{1-C\phi ^2 + C_4}{C\phi ^2+C_4}}$$

I have tried trig substitution to no avail, so I went to look at the slope field on my TI 89, but nothing shows up. It isn't a technical problem, so I was wondering what was going on. Does anyone see anyway I can make further progress? Also, how can I approximate solutions (or just look at them)?