Computer Algebra Recipes

4.1. FIRST-ORDER MODELS 159
PROBLEMS:
Problem 4-3: Nonlinear diode circuit
A linear capacitor with capacitance C is connected in series with a nonlinear
diode that has a current (i){voltage (v) relationoftheformi = av+ bv2 ,with
the coe±cients a and b both positive. The voltage across the capacitor at time
t =0isv = V . Derive the nonlinear ODE governing this circuit. Introducing
the dimensionless variables y = v(t)=V , ¿ = at=C,and¯ = bV=a,rewritethe
ODE in dimensionless form. Analytically solve this ODE, demonstrating that
Maple recognizes the ODE as a Bernoulli equation. For ¯ = 2, plot the solution
overthetimeinterval¿ =0to2.
Problem 4-4: Nonlinear diode revisited
Suppose that the current{voltage relation for the nonlinear diode in the previous
problem is given by the more general relation i = av+bvn ,wheren =2,3,4,5,
::: Derive the corresponding general dimensionless ODE and analytically solve
it for arbitrary n. For the same parameters and time range as in the previous
problem, produce a single plot that shows the solutions for n =2to5. Discuss
the e®ect of increasing n.
Problem 4-5: A potpourri of Bernoulli equations
For each of the following nonlinear ODEs (with prime indicating an x-derivative):
² con¯rm that it is of the Bernoulli type;
² analytically solve the ODE for y(0) = 1;
² plot the solution y(x) overtherangex =0to5.
(a) y 3 y 0 + x ¡1 y 4 = x; (b) y 0 + y = xy 3 ; (c) y ¡ y 0 =3y 3 e ¡2 x ;
(d) y 3 +3y 2 y 0 =4.
Problem 4-6: General solution
Determine the general solution of the following Bernoulli equation:
xy 0 + y = x 3 y 6 :
Problem 4-7: Laser beam competition
In the theory of stimulated thermal scattering [BEP71], the intensities IL and IS
of two interacting collinear laser beams traveling in the z-direction are governed
by the following pair of coupled ¯rst-order nonlinear ODEs:
dIL
dz = ¡gILIS ¡ ®IL; dIS
dz = gILIS ¡ ®IS;
where the gain coe±cient g>0andtheabsorption coe±cient ® ¸ 0.
(a) By adding the two equations and eliminating IL, show that a single
Bernoulli equation can be obtained for the intensity IS.
(b) Analytically solve this Bernoulli equation for IS(z).
(c) Plot ln(IS(z)=IS(0)) for z =0to10cmgiventhatIS(0)=IL(0) = 0:01,
gIL(0) = 1 cm ¡1 ,and(a)® =0,(b)® =0:5 cm ¡1 . Discuss the results.