unbound solution

**saravananbs** · May 13th 2012, 05:24 AM

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.

**Prove It** · May 13th 2012, 05:29 AM

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?

**saravananbs** · May 13th 2012, 05:31 AM

p is a constant.

**Prove It** · May 13th 2012, 05:52 AM

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

**saravananbs** · May 13th 2012, 05:56 AM

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

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