Solve ihbarfrac{partialpsi_{1}}{partialt}=mc^2psi_{1}

$$i \hbar \frac { \partial \psi _ { 1 } } { \partial t } = m c ^ { 2 } \psi _ { 1 }$$

Solve for c

$$\left\{\begin{matrix}\\c=0\text{, }&\text{unconditionally}\\c\in \mathrm{C}\text{, }&\psi _{1}=0\text{ or }m=0\end{matrix}\right.$$

Steps by Finding Square Root

Swap sides so that all variable terms are on the left hand side.

$$mc^{2}\psi _{1}=iℏ\frac{\mathrm{d}(\psi _{1})}{\mathrm{d}t}$$

Dividing by m\psi _{1} undoes the multiplication by m\psi _{1}.

$$c^{2}=\frac{0}{m\psi _{1}}$$

Divide 0 by m\psi _{1}.

$$c^{2}=0$$

Take the square root of both sides of the equation.

$$c=0 c=0$$

The equation is now solved. Solutions are the same.

$$c=0$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$mc^{2}\psi _{1}=iℏ\frac{\mathrm{d}(\psi _{1})}{\mathrm{d}t}$$

Subtract iℏ\frac{\mathrm{d}(\psi _{1})}{\mathrm{d}t} from both sides.

$$mc^{2}\psi _{1}-iℏ\frac{\mathrm{d}(\psi _{1})}{\mathrm{d}t}=0$$

Reorder the terms.

$$-iℏ\frac{\mathrm{d}(\psi _{1})}{\mathrm{d}t}+m\psi _{1}c^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$m\psi _{1}c^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute m\psi _{1} for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{0±\sqrt{0^{2}}}{2m\psi _{1}}$$

Take the square root of 0^{2}.

$$c=\frac{0±0}{2m\psi _{1}}$$

Multiply 2 times m\psi _{1}.

$$c=\frac{0}{2m\psi _{1}}$$

Divide 0 by 2m\psi _{1}.

$$c=0$$

Solve for m

$$\left\{\begin{matrix}\\m=0\text{, }&\text{unconditionally}\\m\in \mathrm{C}\text{, }&\psi _{1}=0\text{ or }c=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$mc^{2}\psi _{1}=iℏ\frac{\mathrm{d}(\psi _{1})}{\mathrm{d}t}$$

The equation is in standard form.

$$\psi _{1}c^{2}m=0$$

Divide 0 by c^{2}\psi _{1}.

$$m=0$$