978-3-319-17852-3

1.5 One-Dimensional Dynamical Systems 77
4. Find the largest interval where the solution to the IVP exists:
a) t(t − 5)x ′ + x = e −t , x(2) = 1. b) x ′ + 1
t − 3 x = 1 , x(4) = 1.
t − 7
5. Find the regions in the xt plane where the hypotheses of Theorem 1.38
hold:
a) x ′ = 2+t2
3x − x 3 . b) (2t +5x)x′ = t − x.
6. In each of the following problems, find how the solution depends on the
initial condition x(0) = x 0 :
a) x ′ = − 4t
x . b) x′ + x 3 =0. (c) x ′ = x 2 .
7. Verify that the initial value problem x ′ = √ x, x(0) = 0, has infinitely many
solutions of the form
{ 0, t ≤ a
x(t) = 1
4 (t − a)2 , t > a,
where a>0 is fixed. Sketch these solutions for three different values of a.
Why might you not be surprised at this result?
8. Verify that the linear initial value problem
x ′ =
2(x − 1)
, x(0) = 1,
t
has a continuously differentiable solution (i.e., a solution whose first derivative
is continuous) given by
{ at
x(t) =
2 +1, t<0,
bt 2 +1, t > 0,
for any constants a and b. Yet, there is no solution if x(0) ≠1. Do these
facts contradict Theorem 1.38?