Recursive Solution of Differential Equation

Question

I have a sequence of differential equations, which can be solved recursively, i.e, the solution of the first equation is necessary to solve the second equation.

Consider for example

$\frac{\partial}{\partial s}(\frac{1}{\epsilon^2}I_{-2}+\frac{1}{\epsilon} I_{-1} +I_0)=\frac{1}{\epsilon^2}f_0I_{-2}+\frac{1}{\epsilon}(f_0I_{-1}+f_1I_{-2})+(I_{-1}f_0+I_0f_1)$

Where $f_0\; \text{and}\; f_1$ are provided. The differential equations we will get from here are

$\frac{\partial}{\partial s}I_{-2}=f_0I_{-2}$

$\frac{\partial}{\partial s}I_{-1}=f_0I_{-1}+f_1I_{-2}$

and so on.

Is there any hope for solving these type of problem in mathematica? I could not try anything so far.

Answer 1

This is not a recursive system of equations. It is a triangular one. Supposing we have to consider initial conditions as I2[0], I1[0], I0[0] and using the Laplace transform

sol = Solve[{LaplaceTransform[D[1/e^2 I2[s] + 1/e I1[s] + I0[s], s] - 1/e (f0 I1[s] + f1 I2[s]) - (f0 I1[s] + f1 I0[s]), s, z] == 0,
             LaplaceTransform[D[I2[s], s] - f0 I2[s], s, z] == 0,
             LaplaceTransform[D[I1[s], s] - f0 I1[s] - f1 I2[s], s, z] == 0},    
            {LaplaceTransform[I2[s], s, z], 
             LaplaceTransform[I1[s], s, z], 
             LaplaceTransform[I0[s], s, z]}][[1]]

InverseLaplaceTransform[{
             LaplaceTransform[I2[s], s, z], 
             LaplaceTransform[I1[s], s, z], 
             LaplaceTransform[I0[s], s, z]} /. sol, z, s]

Answer 2

I changed the variables from {I[-2],I[-1],I[0]} to {y[0],y[1],y[2]}, mainly to avoid having to deal with minus signs in the variable names. If you only have the three terms to deal with:

y0 = 
 First[y[0][t] /. 
   DSolve[Derivative[1][y[0]][t] == f[0] y[0][t], y[0][t], t]]

Out[10]= E^(t f[0]) C[1]

y1 = 
 First[y[1][t] /. 
   DSolve[Derivative[1][y[1]][t] == f[0] y[1][t] + f[1] y0, y[1][t], 
    t]]

Out[11]= E^(t f[0]) C[2] + E^(t f[0]) t C[1] f[1]

And for your missing "and so on" equation, assuming I didn't screw this up (f0 and f1 seem reversed here, as noted in the comment, compared to the previous two equations):

y2 = 
 First[y[2][t] /. 
   DSolve[Derivative[1][y[2]][t] == f[0] y1 + f[1] y[2][t], y[2][t], 
    t]]

Out[13]= 
E^(t f[1]) C[3] + (
 E^(t (f[0] - f[1]) + t f[1])
   f[0] (C[2] (f[0] - f[1]) + C[1] (-1 + t (f[0] - f[1])) f[1]))/(f[
    0] - f[1])^2