Wave packets not satisfying the Schrödinger equation?

The time-independent Schrödinger equation of a free particle in 1 dimension is $$$$-\frac{\hbar^2}{2m}\partial^2_x\psi(x) = E\psi(x)$$$$ which has solutions in form of $$e^{ikx}$$, where $$k=\frac{\sqrt{2mE}}{\hbar}$$ and $$E = \frac{\hbar^2k^2}{2m}$$.

Any superposition of these functions must be also a solution to the Schrödinger-equation, so let say $$\psi(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}dk\phi(k)e^{ikx}.$$

Then we expect that $$\partial_x^2\psi(x) = - \frac{2mE}{\hbar^2}\psi(x)$$ still holds. However, $$\partial_x^2\psi(x) = \partial_x^2\frac{1}{\sqrt{2\pi}}\int\limits_{\infty}^{\infty}dk\phi(k)e^{ikx} \\ = \frac{1}{\sqrt{2\pi}}\int\limits_{\infty}^{\infty}dk \partial_x^2 \phi(k)e^{ikx} \\ = \frac{1}{\sqrt{2\pi}}\int\limits_{\infty}^{\infty}dk \phi(k)(ik)^2e^{ikx}$$.

So the Schrödinger-equation implies that $$\frac{1}{\sqrt{2\pi}}\int\limits_{\infty}^{\infty}dk \phi(k)(ik)^2e^{ikx} = -\frac{2mE}{\hbar^2}\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}dk\phi(k)e^{ikx}$$ That is, $$\int\limits_{\infty}^{\infty}dk k^2\phi(k)e^{ikx} = \frac{2mE}{\hbar^2}\int\limits_{\infty}^{\infty}dk \phi(k)e^{ikx},$$ for any functions $$\phi(k)$$. Does this mean that wave packets do not satisfy the Schrödinger equation?