# unbound solution

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• May 13th 2012, 06:24 AM
saravananbs
unbound solution
for an ordinary differential equation $y" + py' +(p^2/4) y =0$
there exist a unbounded solution y(x)

$limit$ $x->infinite$ $|y(x)/x|$ exist.

can anyone explain how the unbounded solution exist.
• May 13th 2012, 06:29 AM
Prove It
Re: unbound solution
Quote:

Originally Posted by saravananbs
for an ordinary differential equation $y" + py' +(p^2/4) y =0$
there exist a unbounded solution y(x)

$limit$ $x->infinite$ $|y(x)/x|$ exist.

can anyone explain how the unbounded solution exist.

Is p a constant or a function of x?
• May 13th 2012, 06:31 AM
saravananbs
Re: unbound solution
p is a constant.
• May 13th 2012, 06:52 AM
Prove It
Re: unbound solution
Well then the characteristic equation would be \displaystyle \begin{align*} m^2 + p\,m + \frac{p^2}{4} &= 0 \end{align*}. Solving for m gives

\displaystyle \begin{align*} m^2 + p\,m + \frac{p^2}{4} &= 0 \\ \left(m + \frac{p}{2}\right)^2 &= 0 \\ m + \frac{p}{2} &= 0 \\ m &= -\frac{p}{2} \end{align*}

Since this is a repeated root, the solution to your DE is \displaystyle \begin{align*} y = A\,e^{-\frac{p}{2}x} + B\,x\,e^{-\frac{p}{2}x} \end{align*}.

Now you should be able to evaluate \displaystyle \begin{align*} \lim_{x \to \infty}\left|\frac{y}{x}\right| \end{align*}.
• May 13th 2012, 06:56 AM
saravananbs
Re: unbound solution
is the answer is zero. when x-->infinite