Need to separate and integrate dv/dt = v+2t

Question

turingtest · Answer

This differential equation does not look separable to me. Are you sure you typed it correctly?

anonymous · Answer

This equation is not separable.  Rather, the easiest way to solve it is with an integrating factor.

turingtest · Answer

v'-v=2t
integrating factor:e^t
(ve^t)'=2te^tdt
now integrate

jamesj · Answer

No, integrating factor is e^(-t)

turingtest · Answer

my bad...

jamesj · Answer

MIT should pay me a commission ... but @jd50, if this method is obscure, watch this solid lecture: http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-3-solving-first-order-linear-odes/

turingtest · Answer

v'-v=2t
multiply by the integrating factor:e^(-t)
[(ve^(-t)]'=2te^(-t)dt
now integrate

turingtest · Answer

I learned my technique from this site: http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

myininaya · Answer

$$v'-v=2t$$
multiply both sides by y(x)>0 such that
we have
$$yv'-yv=y \cdot 2t$$
we want to write the right hand side as (vy)'
to do that we need y'=-y
=> dy/dt=-y
=>-1/y dy=dt
integrating both sides we get
-ln(y)=t+C
we choose C=0 
so we have
t=-ln(y)
=> -t=ln(y)=> e^(-t)=y
so we can write
$$(e^{-t}v)'=e^{-t} \cdot 2t$$

myininaya · Answer

oops multiply both sides by y(t) lol