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- Homework Statement
- please see below

- Relevant Equations
- please see below

Part 1

$$\Delta u(x)=\Delta v(|x|)$$

Substitute $$|x|=r=\sqrt{\sum_{i=1}^n{x^2_i}}$$

$$u'(x)= v'(r)\frac{\sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^n{x^2_i}}}$$

$$u''(x)=v''(r)\frac{\sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^n{x^2_i}}}+v'(r)f(x)=v''(r)+v'(r)f(x)$$

$$f(x)=\frac{n\sqrt{\sum_{i=1}^nx_i^2}-\frac{\sum_{i=1}^nx_i \sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^nx_i^2}}}{\sum_{i=1}^nx_i^2}=\frac{nr-(\frac{r^2}{r})}{r^2}=\frac{n-1}{r}$$

$$u''(x)=v''(r)+\frac{n-1}{r}v'(r)$$part 2

make the substitution $$v'=a$$

$$0=a'+\frac{n-1}{r}a$$

$$\frac{a'}{a}=\frac{1-n}{r}$$

$$\frac{d}{da} ln(a)=\frac{1-n}{r}$$

$$ln(a)=\int\frac{1-n}{r}dr$$

$$ln(a)=(1-n)ln(r)+C$$

$$a=Cr^{1-n}$$

$$v'=Cr^{1-n}$$

$$v=Cr^{-n}+D$$

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