- 125

- 11

- Problem 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?

$$ \frac{d^2y}{dr^2} = r^2y$$

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?

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