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Department of Mathematics
Differential Equations
1. Show that the following sequences converge uniformly on the indicated domain
(a) fn (x) =
1 −nx
e
, x ∈ [0, ∞)
n
(b) fn (x) = xn (1 − x), x ∈ [0, 1]
2. Determine which of the following sequences converge uniformly in (-1,1)
(a) fn (x) =
1 − xn
1−x
(b) fn (x) =
nx
1 + n2 x2
(c) fn (x) =
nx
1 + n3 x3
3. Test the uniform convergence of the series
(a)
X
rn sin nx, r ∈ (0, 1)
(b)
X
rn cos nx, r ∈ (0, 1)
(c)
X (−1)n−1
n + x2
4. Let (gn ) be a sequence of differentiable functions defined on [0, 1], and assume that
gn (0) = gn0 (0) for all n. Suppose also that |gn0 (x)| ≤ 1 for all n and all x ∈ [0, 1]. Prove
that there is a subsequence of (gn ) converging uniformly on [0, 1].
5. Suppose that X is compact and let fn : X → R be a sequence of equicontinuous
functions such that fn → f pointwise on X. Show that the convergence is actually
uniform.
6. Let F be the set of continuous functions f : [0, 1] → R satisfying
|f (x) − f (y)| ≤ |x − y| for all 0 ≤ x, y ≤ 1
as well as f (0) = 0. Show that F is compact
7. Find the solution of the initial value problem (IVP) and the maximal interval where
solution is defined:
1
π
y
= 0, y(−2) = 1 (b)y 0 + (sec t)y =
, y( ) = 1.
(a) y 0 + 2
t −1
t−1
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8. Show that there is an infinite family of solution to the problem
t2 y 0 − 2ty = t5 , y(0) = 0,
all of which exists in R. Does this violate Theorem of Uniqueness of solution of IVP.
9. Show that for any differentiable function f (t), t ∈ R.,all solutions of y 0 + y = f + f 0
tends to f (t) as t → +∞.
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10. Show that if p is differentiable function and lim p(t) = +∞, then all solution of y 0 +p0 y =
t→∞
0 tend to zero as t → +∞.
11. (Riccati’s eqution): y 0 = A(t)y 2 + B(t)y + C(t) is called Riccati’s equation.
(a) Show that if f (t) is a solution, then y = f +
equation in v.
1
v
reduces the equation to a linear
(b) Find general solution of y 0 = (1 − t)y 2 + (2t − 1)y − t given a solution f (t) = 1.
12. Solve the following IVP:
(t − 1)y 0 = 2y,
y(1) = 1
Explain the results in view of the theory of existence and uniqueness of IVPs.
13. Show that the Lipschitz condition is satisfied by the function | sin y| + t at every point
on the ty-plane though its partial derivative with respect to y does not exist on the line
y = 0.
14. Check that the local existence and uniqueness theorem applies to the initial value problems
(a) y 0 = t + y 2 ,
y0
(b) e = y,
(c)
y0
y(0) = 0,
y(0) = 1,
= | sin(y)|, y(0) = 0.
15. Show that the solution of y 0 = 2 + sin x, x(0) = 0 cannot vanish for t > 0.
16. Show that the solutions of x0 = sin(tx) are even functions.
17. show that the solution of x0 = t2 x4 + 1, x(0) = 0 is an odd function.
18. Show the problem x0 = max{1, x}, x(0) = 1 has unique solution for all t. Find the
solution.
19. show that the solutions of the following problems exists for all t
(a) x0 = sin x
(b) x0 = ln(1 + x2 )
√
20. Show that the successive approximations for the IVP: x0 = 2t − 2 x+ , x(0) = 0, x+ =
max{x, 0}, does not converge.
Hint: Picard iterations: x2n = 0, x2n+1 = t2 . So the sequence is

0
xn (t) =
t2
n is even,
nis odd.
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21. Show that the solutions of y 0 = sin(y) are defined on all IR
22. Show that the solution y 0 = t2 y 4 + 1, y(0) = 0 is odd function.
23. Show that if f (y) > 0 the solutions of y 0 = f (y) can not be even.
24. Determine if the following IVP has unique solutions. Then find a solution.
t2 y − 4y
y0
(a) y 0 =
, y(0) = 1 (b)
= −1, y(1) = 0 (c) ty 0 = 2(y − 4), y(0) = 0
t+2
y+1
25. Show that y 2 + t2 = 1 is the singular solution of (y 0 )2 = y 2 + t2 − 1.
26. Find singular solution of the following ODEs:
p
(a) y 0 = 1 − y 2 (b)(y 0 )2 − ty 0 + x = 0 (c) (y 0 )2 = 4(y − 1)
(d) y 2 (1 + y 02 ) = 1.