Solve C=Adx/dt+Bx+Ddx^2/dt^2

$$C = A \frac { d x } { d t } + B x + D \frac { d x ^ { 2 } } { d t ^ { 2 } }$$

Solve for A

$$A\in \mathrm{R}$$

$$C=Bx$$

Steps for Solving Linear Equation

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

$$A\frac{\mathrm{d}(x)}{\mathrm{d}t}+Bx+D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}=C$$

Subtract Bx from both sides.

$$A\frac{\mathrm{d}(x)}{\mathrm{d}t}+D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}=C-Bx$$

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

$$A\frac{\mathrm{d}(x)}{\mathrm{d}t}=C-Bx-D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}$$

Reorder the terms.

$$A\frac{\mathrm{d}(x)}{\mathrm{d}t}=-D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}-Bx+C$$

The equation is in standard form.

$$0=C-Bx$$

This is false for any A.

$$A\in$$

Solve for B

$$\left\{\begin{matrix}B=\frac{C}{x}\text{, }&x\neq 0\\B\in \mathrm{R}\text{, }&C=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$A\frac{\mathrm{d}(x)}{\mathrm{d}t}+Bx+D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}=C$$

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

$$Bx+D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}=C-A\frac{\mathrm{d}(x)}{\mathrm{d}t}$$

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

$$Bx=C-A\frac{\mathrm{d}(x)}{\mathrm{d}t}-D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}$$

Reorder the terms.

$$Bx=-D\frac{\mathrm{d}(x^{2})}{\mathrm{d}t^{2}}-A\frac{\mathrm{d}(x)}{\mathrm{d}t}+C$$

The equation is in standard form.

$$xB=C$$

Divide both sides by x.

$$\frac{xB}{x}=\frac{C}{x}$$

Dividing by x undoes the multiplication by x.

$$B=\frac{C}{x}$$