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Can anyone help me with the following differential equation? $$ 2x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0, $$ where $\delta$ represents the Dirac Delta.

I tried Mathematica but with no luck. And I really don't know where to start... Can anyone help me?

Note: I first thought that such an equation wouldn't be solvable but If I search for a solution of $x(t)x''(t) - x'(t)^2 + kx(t)^2\delta(t - a) =0,$ mathematica can actually find a solution so maybe the above equation can also be solved.

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1 Answer 1

up vote 2 down vote accepted

Dividing the whole equation by $x(t) x'(t)$, we have

$$ 2 \frac{x''(t)}{x'(t)} - \frac{x'(t)}{x(t)} + k \frac{\delta(x-a) x(t)}{x'(t)} = 0 $$

hence

$$ \frac{d}{dt}\log\left\{\frac{x'(t)^2}{x(t)}\right\} + k \frac{\delta(x-a) x(t)}{x'(t)} = 0 $$

If you integrate the equation around $a$, you have

$$ \int_{a-\epsilon}^{a+\epsilon} \left(\frac{d}{dt}\log\left\{\frac{x'(t)^2}{x(t)}\right\} + k \frac{\delta(x-a) x(t)}{x'(t)}\right)dt = \log\left\{\frac{x'(t)^2}{x(t)}\right\}_{a-\epsilon}^{a+\epsilon} + k \frac{x(a)}{x'(a)} $$

Taking the limit as $\epsilon \to 0$,

$$ \left[\log\left\{\frac{x'(t)^2}{x(t)}\right\}\right]_{t=a} + k \frac{x(a)}{x'(a)} = 0, $$

where

$$ [f(t)]_{t=a} = \lim_{t \to a^+} f(t) - \lim_{t \to a^-} f(t) $$

is the jump condition in $t=a$.

Now, you have the problem

$$ \frac{d}{dt}\log\left\{\frac{x'(t)^2}{x(t)}\right\} = 0 \qquad \begin{cases}t < a\\ t > a\end{cases} $$

with

$$ \left[\log\left\{\frac{x'(t)^2}{x(t)}\right\}\right]_{t=a} + k \frac{x(a)}{x'(a)} = 0, $$

which can be easily integrated.

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1  
Hum, I'm having problems understanding the first passage. Shouldn't: $2~x''/x'-x'/x$ be equal to $\frac{d}{dt}\log(x'^2/x)$? –  PML May 6 '13 at 22:44
    
True. I've edited accordingly. –  Pragabhava May 7 '13 at 0:00
    
Thank you very much, truly, by your very complete answer. Thank not only in answering this question but now I already know where to start to solve similar differential equations.Thank you –  PML May 7 '13 at 2:01
1  
No problem. If you are really interested on what's behind delta functions in ode's, I recommend you to read the chapter on Green's function from Friedman's Principles and Techniques of Applied Mathematics. It is a fantastic introduction to the linear case, and it will provide great insight for nonlinear problems. –  Pragabhava May 7 '13 at 3:47
    
I know the book, I've been reading it in my spare time but haven't gotten to the Dirac delta chapter. Fantastic book, indeed. Thank you for the advice. –  PML May 7 '13 at 9:46

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