I have been solving this question, and I proved n=1, assumed n=k is correct and attempted to solve n=k+1. I got to the point where $$\frac{(d^{k+1}y)}{dx^{k+1}} = \frac{d⋅d^ky}{dx⋅dx^k}$$. Although I got this, I don't know how I can combine the n=k equation and $$\frac{d}{dx}$$. Can someone help me solve this problem?
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You have $\frac{d}{dx}\left(\frac{d^k y}{dx^k}\right)$. You could try substituting your inductive hypothesis for $\frac{d^k y}{dx^k}$ and proceed from there. So you will be doing a differentiation, namely $``\frac{d}{dx}\left(\text{insert inductive hypothesis for }\frac{d^k y}{dx^k}\text{ here}\right)\text{''}$. – Minus One-Twelfth Apr 20 '19 at 14:37
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$$\frac{d^{k+1}y}{dx^{k+1}} = \frac{d}{dx}\left(\frac{d^ky}{dx^k}\right)$$ $$ = \frac{d}{dx}\left(k3^{k-1}e^{3x} + x3^ke^{3x}\right)$$ $$= k3^ke^{3x} + \left(3^ke^{3x}+x3^{k+1}e^{3x}\right)$$ $$= \left(k + 1 \right)3^ke^{3x} + x3^{k+1}e^{3x}$$ which is the same as the relation for $n = k+1$
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