Hi, welcome to PF.
In induction, you start by *proving* the base case (n=1 in this example), and then *assuming* the case for ##n=k## then *proving* that the proposition holds for ##n=k+1##, then finally *concluding* it holds for all positive integral n.
You shouldn't *assume* the base case, as you wrote ("It is assumed that this is true for n=1"). In this case, the base case has already been established based on the previous result in part a). Just quote that result.
Subsequently, you wrote "assume this is true for n=k+1". This is wrong. You should prove that the assumption that it holds for n=k implies that it holds for n=k+1. You actually did this, but writing the assumption before the proof is wrong. It is acceptable to start this off by stating: "We then have to show that ##\displaystyle \frac{d^{2(k+1)}y}{dx^{2(k+1)}} = x\sinh x + 2(k+1)\cosh x##", so it is clear what you're hoping to end up with. But you shouldn't state it as an assumption before proving it. Semantics, but these are important in a proof.
A minor point: you did a bit of unnecessary work in your derivation by going through the odd derivatives of the function. E.g. to get the fourth derivative, you calculated the third derivative. This is not necessary. You can just apply the second derivative directly based on your previous result, getting ##\displaystyle \frac{d^4y}{dx^4} = \frac{d^2}{dx^2}(\frac{d^2y}{dx^2}) = \frac{d^2}{dx^2}(x\sinh x + 2\cosh x) = x\sinh x + 2\cosh x + 2\cosh x = x\sinh x + 4\cosh x##. The same applies in your inductive step. Simplifies your working quite a bit.
