Introductory Differential Equations using Sage - William Stein

1.2. INITIAL VALUE PROBLEMS 15
k1*eˆt + k2*eˆ(-t)
sage: P = plot(eˆt/2-eˆ(-t)/2,0,3)
sage: show(P)
Figure 1.4: Solution to IVP x ′′ − x = 0, x(0) = 0, x ′ (0) = 1.
You see in Figure 1.4 that the solution tends to infinity, as t gets larger.
Exercises:
1. Find the value of the constant C that makes x = Ce 3t a solution to the IVP x ′ = 3x,
x(0) = 4.
2. Verify that x = (C+t)cos(t) satisfies the differential equation x ′ +xtan(t)−cos(t) = 0
and find the value of C that gives the initial condition x(2π) = 0.
3. Determine a value of the constant C so that the given solution of the differential
equation satisfies the initial condition.
(a) y = ln(x + C) solves e y y ′ = 1, y(0) = 1.
(b) y = Ce −x + x − 1 solves y ′ = x − y, y(0) = 3.
4. Use Sage to check that the general solution to the falling body problem
mv ′ + kv = mg,
is v(t) = mg
k
+ ce−kt/m . If v(0) = v 0 , you can solve for c in terms of v 0 to get
c = v 0 − mg
k . Take m = k = v 0 = 1, g = 9.8 and use Sage to plot v(t) for 0 < t < 1.