MATH 395 PROBLEMS 5

IGOR KRIZ

Regular problems:

1. Find all solutions to the differential equations

(a) y′ = yx^{3}

(b) y′ = (x + 2y)^{2}

(c) y′ = sin(x)y + cos(x).

2. Prove that if (X, ρ) is a metric space, then another metric d on X can be defined by d(x, y) = max{1, ρ(x, y)}. Prove that the identity is a homeomorphism of (X, d) to (X, ρ). Prove also that (X, ρ) is complete if and only if (X, d) is complete.

3. Prove that there does not exist a differential equation

y^{(n)} = f(x, y, y′, …, y^{(n−1)})

with f continuous and Lipschitz in the last n variables which would have the solutions

y = ax^{2} + (4a + 2)x + a + 3.

for all a ∈ ℝ. [Use the existence and uniqueness theorem.]

4. Define ℝ^{∞}as the set of all infinite sequences

x_{1}, x_{2}, x_{3}, …

of real numbers. Define

ρ((x_{n}), (y_{n})) = ∑_{i=1}^{∞} \frac{1}{2^{i}} max\{1, |x_{i} − y_{i}|\}.

(see Problem 2). Is ℝ^{∞} a metric space? Is it complete?

Challenge problems:

5. Let f(x, y) be a continuous function with continuous partial derivative by y such that f(x_{0}, y_{0}) = 0, and

\frac{∂f}{∂y} (x_{0}, y_{0}) ≠ 0.

Prove that there exists an ϵ > 0 and a unique function

g : (x_{0} − ϵ, x_{0} + ϵ) → ℝ

such that

f(x, g(x)) = 0.

[Differentiate by x and, using the chain rule, write a differential equation for y = g(x).]

6. Find a differential equation whose characteristics (=solutions defined on a maximal possible interval) have domains

(a) (a, +∞), a ∈ ℝ

(b) (−a, a), a > 0.