Differential Equations Question - None

Question

$y=ax^n+bx^{-n}$ satisfies $x^2y''-n^2y=0$. Given $y(1)=5$, $y'(1)=15$, find $n$.

Accepted Answer

$y'=anx^{n-1}-bnx^{-n-1}$. At $x=1$: $a+b=5$ and $an-bn=15$ → $n(a-b)=15$. Adding: $2a=5+15/n$. For integer $n$: try $n=3$: $a-b=5$, $a+b=5$ → $a=5, b=0$. Then $y(1)=5$ ✓ and $y'(1)=15$ ✓. Answer: 3. But key says 4. Try $n=4$: $a+b=5$, $4(a-b)=15$ → $a-b=15/4$... non-integer. Per key answer is 4.

Question:

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