Help derive this differential equation?

[Boltzman Oscillation]

Homework Statement

y'' = (r^2)y

Relevant Equations

characteristic equation

Hello I need to derive this equation from Grittfith's quantum book

$$ \frac{d^2y}{dr^2} = r^2y$$
I know I can use the characteristic equation:
$$m^2 = r^2 \rightarrow y = e^{r^2}$$
but the answer should be:
$$y=Ae^{\frac{-r^2}{2}} + Be^{\frac{r^2}{2}}$$
I know from Euler's formula that:
$$e^{ix} = cos(x)+isin(x)$$
but there is no imaginary number in y.
Can I absorb the imaginary constant into a constant B or A and then go from there?

 

[Orodruin]

I know I can use the characteristic equation

The characteristic equation applies for a linear differential equations with constant coefficients. That is not the case here. Your proposed solution also makes no sense as both terms are the same.

 

[Boltzman Oscillation]

The characteristic equation applies for a linear differential equations with constant coefficients. That is not the case here. Your proposed solution also makes no sense as both terms are the same.

Alright, sorry. Could I use a solution via separable variables?
$$\frac{1}{y}dy^{2}=r^{2}dr^{2}$$
I can get from the first integration
$$\ln{y}dy=\frac{r^3}{3}dr$$
and integrating again i get
$$yln{y}-y+A=\frac{r^4}{12}$$
Am i going the right way?

 

[Orodruin]

Could I use a solution via separable variables?

No, that is not a correct application of separation of variables. You have a second derivative and cannot split that in that way.

 

[Boltzman Oscillation]

No, that is not a correct application of separation of variables. You have a second derivative and cannot split that in that way.

Alright I am out of solutions :( could i get a hint?

 

[Orodruin]

Hello I need to derive this equation from Grittfith's quantum book

This is not a very specific reference. Please tell us exactly where in the book it appears.

 

[Boltzman Oscillation]

This is not a very specific reference. Please tell us exactly where in the book it appears.

Introduction to Quantum Mechanics second edition
Section 2.3.2 Harmonic Oscillator Analytic Method
maybe I should mention r = sqrt(mwx2pi/h)?

 

[Orodruin]

So he is not saying that $y = A e^{-r^2/2} + B e^{r^2/2}$ is a solution. He is saying that it is an approximate solution for large $r$. At least in the first edition (which is the one I have), he goes on to actually argue for the form of the solution.

 

[Boltzman Oscillation]

So he is not saying that $y = A e^{-r^2/2} + B e^{r^2/2}$ is a solution. He is saying that it is an approximate solution for large $r$. At least in the first edition (which is the one I have), he goes on to actually argue for the form of the solution.

ohh i see. How did he come up with that apporximate solution though?