Solve d^2g/dt^2=-adg/dt-bdh/dt+fde/dt

$$\frac { d ^ { 2 } g } { d t ^ { 2 } } = - a \frac { d g } { d t } - b \frac { d h } { d t } + f \frac { d e } { d t }$$

Solve for a

$$a\in \mathrm{R}$$

Steps for Solving Linear Equation

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

$$\left(-a\right)\frac{\mathrm{d}(g)}{\mathrm{d}t}-b\frac{\mathrm{d}(h)}{\mathrm{d}t}+f\frac{\mathrm{d}(e)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}$$

Subtract f\frac{\mathrm{d}(e)}{\mathrm{d}t} from both sides.

$$\left(-a\right)\frac{\mathrm{d}(g)}{\mathrm{d}t}-b\frac{\mathrm{d}(h)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}-f\frac{\mathrm{d}(e)}{\mathrm{d}t}$$

Add b\frac{\mathrm{d}(h)}{\mathrm{d}t} to both sides.

$$\left(-a\right)\frac{\mathrm{d}(g)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}-f\frac{\mathrm{d}(e)}{\mathrm{d}t}+b\frac{\mathrm{d}(h)}{\mathrm{d}t}$$

Reorder the terms.

$$-a\frac{\mathrm{d}(g)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}+b\frac{\mathrm{d}(h)}{\mathrm{d}t}-f\frac{\mathrm{d}(e)}{\mathrm{d}t}$$

The equation is in standard form.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Solve for b

$$b\in \mathrm{R}$$

Steps for Solving Linear Equation

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

$$\left(-a\right)\frac{\mathrm{d}(g)}{\mathrm{d}t}-b\frac{\mathrm{d}(h)}{\mathrm{d}t}+f\frac{\mathrm{d}(e)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}$$

Subtract f\frac{\mathrm{d}(e)}{\mathrm{d}t} from both sides.

$$\left(-a\right)\frac{\mathrm{d}(g)}{\mathrm{d}t}-b\frac{\mathrm{d}(h)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}-f\frac{\mathrm{d}(e)}{\mathrm{d}t}$$

Add a\frac{\mathrm{d}(g)}{\mathrm{d}t} to both sides.

$$-b\frac{\mathrm{d}(h)}{\mathrm{d}t}=\frac{\mathrm{d}(g)}{\mathrm{d}t^{2}}-f\frac{\mathrm{d}(e)}{\mathrm{d}t}+a\frac{\mathrm{d}(g)}{\mathrm{d}t}$$

The equation is in standard form.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$