Existence and Uniqueness of Solutions

Conditions that guarantee exactly one solution curve through a given point, why such curves can never cross, and what goes wrong when the conditions fail.

The ODE Project, Existence and Uniqueness of Solutions

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13 exercises

Exercise 1

Which of the following initial value problems are guaranteed to have a unique solution by the Existence and Uniqueness Theorem (Existence and Uniqueness Theorem)? In each case, justify your conclusion.

$y' = y^2 + y^3$ , $y(0) = 1$
$y' = \sqrt[4]{y}$ , $y(1) = 0$
$y' = \sqrt[4]{y}$ , $y(1) = 1$
$x' = \dfrac{t}{x^2 - 4}$ , $x(0) = 2$
$x' = \dfrac{t}{x^2 - 4}$ , $x(2) = 0$
$y' = x \sin y$ , $y(0) = 0$
$y' = \dfrac{1}{t - 1} y + 2t$ , $y(1) = 1$

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Exercise 2

Find an explicit solution to the initial value problem Use your solution to determine the interval of existence.

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Exercise 3

Consider the initial value problem

Show that the constant function, $y(t) \equiv 0$ , is a solution to the initial value problem.
Show that is a solution for the initial value problem, where $t_0$ is any real number. Hence, there exists an infinite number of solutions to the initial value problem.
Explain why this example does not contradict the Existence and Uniqueness Theorem.

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Exercise 4

Consider the initial value problem

Use the fact that $y' = 2ty + t$ is a first-order linear differential equation to find a solution to the initial value problem.
Let $\phi_0(t) = 1$ and use Picard iteration to find $\phi_n(t)$ .
Show that the sequence $\{ \phi_n(t) \}$ converges to the exact solution that you found in part (a) as $n \to \infty$ .

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Exercise 5

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Use the Fundamental Theorem of Calculus to show that the function $u = u(t)$ is a solution to the initial value problem if and only if $u$ is a solution to the integral equation

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Exercise 6

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

If $\partial f/ \partial x$ is continuous on the rectangle prove that there exists a $K \gt 0$ such that for all $(t, x_1)$ and $(t, x_2)$ in $R$ .

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Exercise 7

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Define the sequence $\{ u_n \}$ by Use the result of the previous exercise to show that

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Exercise 8

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Show that there exists an $M \gt 0$ such that

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Exercise 10

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Use mathematical induction to show that

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Exercise 11

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Since we can view $u_n(t)$ as a partial sum for the series If we can show that this series converges absolutely, then our sequence will converge to a function $u(t)$ . Show that where $h$ is the maximum distance between $(t_0, x_0)$ and the boundary of the rectangle $R$ . Since $|u_n(t) - u_{n -1}(t)| \to 0$ , we know that…

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Exercise 12

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

To show uniqueness, assume that $u(t)$ and $v(t)$ are both solutions to Show that

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Exercise 13

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Define then $\phi(t_0) = 0$ and $\phi(t) \geq 0$ for $t \geq t_0$ . Show that
Since we know that Use this fact to show that Conclude that for $t \geq t_0$ or for all $t \geq t_0$ and $u(t) = v(t)$ .

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Exercise 14

Let $\phi_n(x) = x^n$ for $0 \leq x \leq 1$ and show that This is an example of a sequence of continuous functions that does not converge to a continuous function, which helps explain the need for uniform continuity in the proof of the Existence and Uniqueness Theorem.

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