m g - K \frac { d x } { d t } = m \frac { d ^ { 2 } x } { d t ^ { 2 } }
Solve for K (complex solution)
K\in \mathrm{C}
g=0\text{ or }m=0
Solve for g (complex solution)
\left\{\begin{matrix}\\g=0\text{, }&\text{unconditionally}\\g\in \mathrm{C}\text{, }&m=0\end{matrix}\right.
Solve for K
K\in \mathrm{R}
g=0\text{ or }m=0
Solve for g
\left\{\begin{matrix}\\g=0\text{, }&\text{unconditionally}\\g\in \mathrm{R}\text{, }&m=0\end{matrix}\right.
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-K\frac{\mathrm{d}(x)}{\mathrm{d}t}=m\frac{\mathrm{d}(x)}{\mathrm{d}t^{2}}-mg
Subtract mg from both sides.
0=-gm
The equation is in standard form.
K\in
This is false for any K.
mg=m\frac{\mathrm{d}(x)}{\mathrm{d}t^{2}}+K\frac{\mathrm{d}(x)}{\mathrm{d}t}
Add K\frac{\mathrm{d}(x)}{\mathrm{d}t} to both sides.
mg=0
The equation is in standard form.
g=0
Divide 0 by m.
-K\frac{\mathrm{d}(x)}{\mathrm{d}t}=m\frac{\mathrm{d}(x)}{\mathrm{d}t^{2}}-mg
Subtract mg from both sides.
0=-gm
The equation is in standard form.
K\in
This is false for any K.
mg=m\frac{\mathrm{d}(x)}{\mathrm{d}t^{2}}+K\frac{\mathrm{d}(x)}{\mathrm{d}t}
Add K\frac{\mathrm{d}(x)}{\mathrm{d}t} to both sides.
mg=0
The equation is in standard form.
g=0
Divide 0 by m.
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