Solve md^2y^2/dt^2+cdx/dt+kx=0

$$m \frac { d ^ { 2 } y ^ { 2 } } { d t ^ { 2 } } + c \frac { d x } { d t } + k x = 0$$

Solve for c

$$c\in \mathrm{R}$$

$$x=0\text{ or }k=0$$

Steps for Solving Linear Equation

Subtract m\frac{\mathrm{d}(y^{2})}{\mathrm{d}t^{2}} from both sides. Anything subtracted from zero gives its negation.

$$c\frac{\mathrm{d}(x)}{\mathrm{d}t}+kx=-m\frac{\mathrm{d}(y^{2})}{\mathrm{d}t^{2}}$$

Subtract kx from both sides.

$$c\frac{\mathrm{d}(x)}{\mathrm{d}t}=-m\frac{\mathrm{d}(y^{2})}{\mathrm{d}t^{2}}-kx$$

The equation is in standard form.

$$0=-kx$$

This is false for any c.

$$c\in$$

Solve for k

$$\left\{\begin{matrix}\\k=0\text{, }&\text{unconditionally}\\k\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m\frac{\mathrm{d}(y^{2})}{\mathrm{d}t^{2}} from both sides. Anything subtracted from zero gives its negation.

$$c\frac{\mathrm{d}(x)}{\mathrm{d}t}+kx=-m\frac{\mathrm{d}(y^{2})}{\mathrm{d}t^{2}}$$

Subtract c\frac{\mathrm{d}(x)}{\mathrm{d}t} from both sides.

$$kx=-m\frac{\mathrm{d}(y^{2})}{\mathrm{d}t^{2}}-c\frac{\mathrm{d}(x)}{\mathrm{d}t}$$

The equation is in standard form.

$$xk=0$$

Divide 0 by x.

$$k=0$$