Solve 1/psi_{varphi}frac{partial^2psi_{phi}}{partialphi^2}=-m^2

$$\frac { 1 } { \psi _ { \varphi } } \frac { \partial ^ { 2 } \psi _ { \phi } } { \partial \phi ^ { 2 } } = - m ^ { 2 }$$

Solve for ψ_φ

$$\psi _{φ}\neq 0$$

$$m=0\text{ and }\psi _{φ}\neq 0$$

Steps for Solving Linear Equation

Variable \psi _{φ} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by \psi _{φ}.

$$1\frac{\mathrm{d}(\psi _{ϕ})}{\mathrm{d}ϕ^{2}}=-\psi _{φ}m^{2}$$

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

$$-\psi _{φ}m^{2}=1\frac{\mathrm{d}(\psi _{ϕ})}{\mathrm{d}ϕ^{2}}$$

Reorder the terms.

$$-\psi _{φ}m^{2}=\frac{\mathrm{d}(\psi _{ϕ})}{\mathrm{d}ϕ^{2}}$$

The equation is in standard form.

$$\left(-m^{2}\right)\psi _{φ}=0$$

Divide 0 by -m^{2}.

$$\psi _{φ}=0$$

Variable \psi _{φ} cannot be equal to 0.

$$\psi _{φ}\in \emptyset$$

Solve for m

$$m=0$$

$$\psi _{φ}\neq 0$$