Laplace transform for set of differential equations

[aaaa202]

I have a set of differential equations with the basic form:

dy_n/dt = t*(a_(n-1)*y_(n-1)+b(n+1)*y_(n+1)-2c_n*y_n)

So the time depence is a simple factor in front of the coefficient matrix. Does this set of differential equations have closed form solutions?

 

[pasmith]

I have a set of differential equations with the basic form:

$$\frac{dy_n}{dt} = t(a_{n-1}y_{n-1}+b_{n+1}y_{n+1}-2c_ny_n)$$

So the time depence is a simple factor in front of the coefficient matrix. Does this set of differential equations have closed form solutions?

Change the independent variable to $x$, where $$\frac{dy_n}{dt} = t\frac{dy_n}{dx}.$$ The resulting system is linear with constant coefficients and therefore has a closed form solution.

 

[aaaa202]

Nice! So I should substitute x=½t²?
Will this work for any system where the time dependence may be pulled outside the parenthesis on the RHS like in my example?
Also if I get a system with constant coefficients I may obtain a steady state solution where dy_n/dx=0. That does not seem realistic considering i started off with a system with time dependent coefficients.