Stochastic differential equation problem

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The discussion revolves around a stochastic differential equation (SDE) represented as dv/dt = -αv + λF + η, where α, λ, and F are constants, v is a variable representing speed, and η is a random value. The poster seeks assistance in analytically solving this equation, noting its similarity to Brownian motion with an applied field. They plan to compare the analytical solution with a numerical one. A change of variable is suggested to transform the equation into a standard form of dv/dt = kv + noise. A link to a resource on solving SDEs is provided for further guidance.
johnt447
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Sorry if this is in the wrong section but i have a problem, I have no experience with stochastic equations well analytically anyway.

The equation i have is the following;

\frac{dv}{dt} = - \alpha v+ \lambda F+\eta

Where alpha lambda and F are constants, v is a variable (speed in this case) and eta is a random value. I believe this is similar to Brownian motion with an applied field, although i have no idea how to solve this analytically i plan to solve it analytically and compare it to a numerical solution. So any help will be most appreciated!
 
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This will turn into a standard equation of the type dv/dt=kv+ noise after a change of variable. For some general methods for solving SDEs, I hope the following link will be of much help -
http://math.berkeley.edu/~evans/SDE.course.pdf
 

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