Question

How would you solve this DE analytically: dx/dt = y(t) - ax(t)
related to this, i've tried to solve this equation, but i'm stuck at this: integral(dx.dt/x)

Answer 1

eh, \(x'=y-ax\)? What exactly is \(y\)? what about \(a\) (I presume a constant)?

Answer 2

P.S. \(\dfrac1x\dfrac{dx}{dt}=\dfrac{d}{dt}\left(\log x\right)\) -- that's just the chain rule

Answer 3

Anyways, this is just a first-order linear differential equation:$$x'+ax=y$$You want to multiply by some integration factor \(\mu\) s.t. \(\mu x'+a\mu x=(\mu x)'=\mu x'+\mu' x\), i.e. \(a\mu=\mu'\). Solving this equation gives us \(\mu=e^{at}\) so we observe:$$e^{at}x'+ae^{at}x=e^{at}y\\(e^{at}x)'=e^{at}y\\e^{at}x=\int e^{at} y\,dt\\x=\frac1{e^{at}}\int e^{at}y\,dt$$

Answer 4

http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

Answer 5

thanks, and sorry for being late, i lost my internet connection
y is a function of t as well as x, and a is constant

Answer 6

You gave \(y\) as a function of \(t\) so I solved assuming that above. \(y(x,t)\) is just \(y(x(t),t)\) though hence it's still just a function of \(t\) since \(x\) itself is a function of \(t\)

Answer 7

you mean you assumed that y is a function of x and t (while it is really not, but makes no difference) s.t. you can solve it?

Answer 8

btw, my second question was this: \[\int\limits \frac{ dx }{ x } dt\]

Answer 9

@amnsbr no I assumed \(y(t)\)

Answer 10

also $$\int\frac{dx}xdt$$ makes no sense

Answer 11

tnx, got it