Solve dx^2/dt^2+12dx/dt+13x=2dx/dt

$$\frac { d x ^ { 2 } } { d t ^ { 2 } } + \frac { 12 d x } { d t } + 13 x = 2 \frac { d x } { d t }$$

Solve for d

$$d\neq 0$$

$$t=-\frac{12}{13}\text{ or }\left(x=0\text{ and }t\neq 0\right)$$

Steps for Solving Linear Equation

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by dt.

$$dt\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}+12dx+13xdt=2\frac{\mathrm{d}(x)}{\mathrm{d}t}dt$$

Subtract 2\frac{\mathrm{d}(x)}{\mathrm{d}t}dt from both sides.

$$dt\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}+12dx+13xdt-2\frac{\mathrm{d}(x)}{\mathrm{d}t}dt=0$$

Combine all terms containing d.

$$\left(t\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}+12x+13xt-2\frac{\mathrm{d}(x)}{\mathrm{d}t}t\right)d=0$$

The equation is in standard form.

$$\left(13tx+12x\right)d=0$$

Divide 0 by 12x+13xt.

$$d=0$$

Variable d cannot be equal to 0.

$$d\in \emptyset$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=-\frac{12}{13}\text{, }&d\neq 0\\t\neq 0\text{, }&x=0\text{ and }d\neq 0\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{12}{13}\text{, }&x\neq 0\text{ and }d\neq 0\\t\neq 0\text{, }&x=0\text{ and }d\neq 0\end{matrix}\right.$$