Why is the de Broglie equation as well as the Schrodinger equation is correct for massive particle?

Starting from special relativity, here I see the de Broglie approximation is valid only if ${m}_{0}=0$

Derivation:

${E}^{2}={P}^{2}{C}^{2}+{m}_{0}^{2}{C}^{4}$. Here we put Plank-Einstein relation $E=h\nu =h\frac{C}{\lambda}$. Finally,

$\lambda =\frac{h}{\sqrt{{P}^{2}+{m}_{0}^{2}{C}^{2}}}\phantom{\rule{2cm}{0ex}}(1)$

If ${m}_{0}=0$ then $\lambda =\frac{h}{p}$ (de Broglie approximation).

Furthermore we know the Schrodinger equation was derived by assuming that the de Broglie approximation is true for all particles, even if ${m}_{0}\ne 0$. But if we take special relativity very strictly then this approximation looks incorrect.

In addition, if we try to derive the Schrodinger equation from the exact relation found in '1', we find completely different equation. For checking it out, lets take a wave function-

$\mathrm{\Psi}=A{e}^{i(\frac{2\pi}{\lambda}x-\omega t)}=A{e}^{i(\frac{\sqrt{{P}^{2}+{m}_{0}^{2}{C}^{2}}}{\hslash}x-\frac{E}{\hslash}t)}\phantom{\rule{2cm}{0ex}}$ (putting $\lambda $ from '1', $\frac{h}{2\pi}=\hslash $ and $E=\hslash \omega $).

Then, $\frac{{\mathrm{\partial}}^{2}\mathrm{\Psi}}{\mathrm{\partial}{x}^{2}}=-\frac{{P}^{2}+{m}_{0}^{2}{C}^{2}}{{\hslash}^{2}}\mathrm{\Psi}=-\frac{{E}^{2}}{{C}^{2}{\hslash}^{2}}\mathrm{\Psi}$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{E}^{2}\mathrm{\Psi}=-{C}^{2}{\hslash}^{2}\frac{{\mathrm{\partial}}^{2}\mathrm{\Psi}}{\mathrm{\partial}{x}^{2}}\phantom{\rule{4cm}{0ex}}(2)$

Again, $\frac{\mathrm{\partial}\mathrm{\Psi}}{\mathrm{\partial}t}=-i\frac{E}{\hslash}\mathrm{\Psi}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}E\mathrm{\Psi}=-i\hslash \frac{\mathrm{\partial}\mathrm{\Psi}}{\mathrm{\partial}t}$

Here we see operator $E=-i\hslash \frac{\mathrm{\partial}}{\mathrm{\partial}t}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{E}^{2}=-{\hslash}^{2}\frac{{\mathrm{\partial}}^{2}}{\mathrm{\partial}{t}^{2}}$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{E}^{2}\mathrm{\Psi}=-{\hslash}^{2}\frac{{\mathrm{\partial}}^{2}\mathrm{\Psi}}{\mathrm{\partial}{t}^{2}}\phantom{\rule{4cm}{0ex}}(3)$

Combining (2) and (3) we find the differential equation:

$\frac{{\mathrm{\partial}}^{2}\mathrm{\Psi}}{\mathrm{\partial}{x}^{2}}=\frac{1}{{C}^{2}}\frac{{\mathrm{\partial}}^{2}\mathrm{\Psi}}{\mathrm{\partial}{t}^{2}}$

It is the Maxwell's equation, not the well known Schrodiner equation!

Therefore for the Schrodinger equation to exist, the de Broglie approximation must hold for ${m}_{0}\ne 0.$ I see a clear contradiction here. Then why is the Schrodinger equation correct after all?