1. ## unbound solution

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

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

can anyone explain how the unbounded solution exist.

2. ## Re: unbound solution

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

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

can anyone explain how the unbounded solution exist.
Is p a constant or a function of x?

3. ## Re: unbound solution

p is a constant.

4. ## Re: unbound solution

Well then the characteristic equation would be \displaystyle \displaystyle \begin{align*} m^2 + p\,m + \frac{p^2}{4} &= 0 \end{align*}. Solving for m gives

\displaystyle \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 \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 \displaystyle \begin{align*} \lim_{x \to \infty}\left|\frac{y}{x}\right| \end{align*}.

5. ## Re: unbound solution

is the answer is zero. when x-->infinite