Is y(x) Identically Zero in This ODE Given Specific Initial Conditions?

[LagrangeEuler]

For ordinary differential equation
$$y''(x)+V(x)y(x)+const y(x)=0$$
for which $\lim_{x \to \pm \infty}=0$ if we have that in some point $x_0$ the following statement is true
$y(x_0)=y'(x_0)=0$ is then function $y(x)=0$ everywhere?

 

[Strum]

I suppose you mean $\lim_{x=\pm \infty} y(x) = 0$? And no the function doesn't have to be $0$ everywhere. An example is $y(x) = \tanh(x)^{2}(1-\tanh(x)^{2})$. (You will have to work out $V(x)$ yourself.)

 

[LagrangeEuler]

Yes $\lim_{x \to \pm \infty}y(x)=0$. Interesting example. Look here

from 2:46 - 4:09.

 

[Strum]

Well $V(x)$ in the above solution is divergent in $0$. The product of $V(x)y(x)$ still exists.