Question

differential equation help

Answer 1

I am suppose to solve the equation for air resistance of a falling body given as

Answer 2

\[m \frac{ dv }{ dt }=(force of gravity) +(force of air resistance)\]

Answer 3

\[m \frac{ dv }{ dt }=mg-\beta v\]

Answer 4

I am not sure where to start with this

Answer 5

I re wrote it as \[m \frac{ dv }{ dt }+betav=mg\] but I do not see where to go from there because I do not seem to see if I need an integrating factor or not

Answer 6

@ikram002p @imqwerty

Answer 7

@jdoe0001 @nincompoop

Answer 8

If memory serves, you have constants \(m,g,\beta\), correct? (mass, acceleration due to gravity, and some drag-like coefficient). The ODE is linear in \(y\), which means you can solve via finding an integrating factor.
\[m \frac{ \mathrm{d}v }{ \mathrm{d}t }=mg-\beta v\iff\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{\beta}{m}v=g\]
The integrating factor thus takes the form of
\[\mu(t)=\exp\left(\int \frac{\beta}{m}\,\mathrm{d}t\right)=e^{\beta t/m}\]
Multiplying both sides of the ODE by \(\mu\), you get
\[e^{\beta t/m}\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{\beta}{m}e^{\beta t/m}v=ge^{\beta t/m}\]
The LHS is a derivative of the product \(\mu(t)v(t)\), so you have
\[\frac{\mathrm{d}}{\mathrm{d}t}\left[e^{\beta t/m}v\right]=ge^{\beta t/m}\implies e^{\beta t/m}v=g\int e^{\beta t/m}\,\mathrm{d}t\]and so on.