How Do You Solve a Nonlinear Differential Equation Involving Exponential Terms?

[CJDW]

Nonlinear DE (with e^t) ?

Good day forum,

I have this wonderful DE :

dx/dt = [a - f '(t)]x + (b + d(c^t))(x^2) - 1

with,
t $$\in$$ [s,T]
x(T) = 0

a, b, d & c are constants.
f(t) = g + h(k^t) , where g, h & k are constants (but I think specifying this is of no importance)

My knowledge of non-linear equations is very limited and would sincerely appreciate any help whatsoever.

CJDW

 

[jackmell]

That looks like a Riccati equation:

$$\frac{dx}{dt}=\left(a-f'(t)\right)x+(b+dc^t)x^2-1$$

$$\frac{dx}{dt}+Q(t)x+R(t)x^2=P(t)=-1$$

and using the standard transformation for a Riccati equation, obtain a second-order (linear) DE:

$$Ru''-(R'-QR)u'-PR^2u=0$$

Now, you can then put the equation in it's Normal form by letting:

$$u=v\text{exp}\left(-1/2\int P dt\right)$$

in order to remove the term involving the first derivative. Yeah, I know this ain't easy. I'm getting this right out of "Intermediate Differential Equations" by Rainville. We then obtain the equation:

$$v''+Iv=0$$

where:

$$I=Q-1/2 P'-1/2 P^2$$

and if I just happens to be a constant, that equation can be easily solved.