Recently, I've met for the first time with the so called "forward/backward" solutions to first order difference/differential equations. It seems that the subject goes back to Blanchard's "Backward and Forward Solutions for Economies with Rational Expectations" https://www.jstor.org/stable/1801627.
Is there some reference with simple examples for this type of calculations? Some textbook or lecture notes?
For example : If I have the differential equation $x' = ax + b$ with $a, b$ real constants I get the following "forward solution" :
$$
x(0) = \left(\frac{b}{a} + x(t)\right)e^{-at} - \frac{b}{a}
$$
and the following "backward solution" :
$$
x(0) = \left(\frac{b}{a} + x(-t)\right)e^{at} - \frac{b}{a}
$$
Is that correct?
I would appreciate any comments or references on the subject
The Crux/Core of your Query is covered by the following Discussion :
A Differential Equation may be solved analytically to get Exact Solutions.
In some cases , the Exact Solutions are either (1) not Possible (theoretically no way to Derive it or Practically too cumbersome to Evaluate or ETC) or (2) not desirable (Discrete values may be wanted or the Solver is looking to get numerical values , not algebraic Equations or ETC)
In such Cases , the technique is to convert the given Differential Equation to a Difference Equation.
Basically , that technique is to substitute Differentials $dy/dx$ with Differences $\Delta y / \Delta x$.
Now , the Differences are (1) APPROXIMATE & (2) come in at least 3 "flavours" :
(1) Take "Next value of $x$" & "Current value of $x$" : $(n+1)(\Delta x)$ & $(n)(\Delta x)$
Thus we have :
$$dy/dx \approx \frac{[y((n+1)(\Delta x))-y((n)(\Delta x))]}{[\Delta x]} \tag{1}$$
Substitute this in the Differential Equation & solve it to get "forward" Solution.
(2) Take "Current value of $x$" & "Previous value of $x$" : $(n)(\Delta x)$ & $(n-)(\Delta x)$
Thus we have :
$$dy/dx \approx \frac{[y((n)(\Delta x))-y((n-1)(\Delta x))]}{[\Delta x]} \tag{2}$$
Substitute this in the Differential Equation & solve it to get "backward" Solution.
(3) Take "Next value of $x$" & "Previous value of $x$" : $(n+1)(\Delta x)$ & $(n-)(\Delta x)$
Thus we have :
$$dy/dx \approx \frac{[y((n+1)(\Delta x))-y((n-1)(\Delta x))]}{[2\Delta x]} \tag{3}$$
Substitute this in the Differential Equation & solve it to get "Central" Solution.
Now , Solutions (1) , (2) , (3) are APPROXIMATE Solutions which may not even match each other.
Here is 1 reference :
http://web.eecs.utk.edu/~mjr/WebAppendices/Q-DiffEqs.pdf
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