Regarding the solutions to this ode

Question

For the ODE $$\dfrac{d^2y}{dx^2}+4x^2y=0$$ I assumed a solution in the form $$y=\sum_{n=0}^{\infty}a_nx^n$$ and I managed to get the following recursive definition: $$a_{n+4}=-\dfrac{4}{(n+3)(n+4)}a_n$$ Given $a_0$, I get one solution $$y_1=a_0\left(1-\dfrac{1}{3}x^4+\dfrac{1}{42}x^8 ...\right)$$ and given $a_1$, I get another solution $$y_2=a_0\left(x-\dfrac{1}{5}x^5+\dfrac{1}{90}x^9 ...\right)$$ I recognise this as a second order ODE, and so I expect only two linearly independent solutions. The thing is, if I was given an $a_2$ or an $a_3$, then don't I get another two linearly independent solutions? This means I have four solutions, which doesn't make any sense. I'm assuming $a_2$ and $a_3$ solutions must therefore be dependent with respect to the ones I listed, but I can't see it.

Answer 1

Good question. If you actually look at the bottom few terms of the power series $\sum_{n \geq 0} a_n x^n$ once you substitute it in, you find $$ 0 = 2a_2 + 6a_3 x + x^2(4a_0+12a_4)+x^3(4a_1+20a_5) + \dotsb, $$ so in fact when one equates coefficients, the $a_2$ and $a_3$ have to vanish anyway: the derivative only lops off the first two terms. This shows the importance of examining the base of the series, as well as the recurrence relation.