Question

How do you solve for x(t):
dx/dt = ct + kx where c and x are constants?
I'm not sure where to move from here because i have both t and x on the RHS please

Answer 1

i think you mean \(c\) and \(k\) are the constants\[\frac{\text dx}{\text dt} = ct + kx\]

Answer 2

Yes, typo there, c and k are constants

Answer 3

Is is allowed to leave x on the right side while you have dx on left?

Answer 4

isn't this a situation where you need to find the integrating factor
\[\rightarrow e^{\int\limits_{}^{}k dt}\]

Answer 5

** sorry that should be a negative k

Answer 6

\[\frac{\text dx}{\text dt} - kx= ct \]

integrating factor is \(R(x)=e^{\int-k\text dx}\)

Answer 7

here is a good explanation....scroll down to examples
http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

Answer 8

Thanks a lot for helping me, had no idea what was hapenning

Answer 9

\[R(x)=e^{-k\int\text dx}=e^{-kx}\]

Answer 10

\[\frac{\text dx}{\text dt} - kx= ct\]

\[\frac{\text d}{\text dt}\left(xe^{-kx}\right) = cte^{-kx}\]
\[xe^{-kx}=c\int te^{-kx}\text dt\]

Answer 11

\[xe^{-kx}=c\frac{ t^2}{2}e^{-kx}+C\]\[x=\frac{ct^2}{2}+Ce^{kx}\]