Short Answer

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dx}{dt} + \cos x = \sin t, x (π) = 0$

The hypotheses of Theorem 1 are satisfied.

The theorem shows that the given initial value problem has a unique solution.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding the partial derivative of the given relation concerning y.

Here, $f (t, x) = sint - cosx$

and

$\begin{array}{l} \frac{\partial f}{\partial x} = - (- \sin x) \\ \frac{\partial f}{\partial x} = \sin x \end{array}$

Step 2: Determining whether Theorem 1 implies the existence of a unique solution or not

Now from Step 1, we find that both of the functions $f (t, x)$ and $\frac{\partial f}{\partial x}$ are continuous in any rectangle containing the point $(π, 0)$ , so the hypotheses of Theorem 1 are satisfied. It then follows from the theorem that the given initial value problem has a unique solution in an interval $t = π$ about of the form $(π - δ, π + δ)$ , where $δ$ is some positive number.

Hence, Theorem 1 implies that the given initial value problem has a unique solution.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dx}{dt} + \cos x = \sin t, x (π) = 0$

Step by Step Solution

Step 1: Finding the partial derivative of the given relation concerning y.

Step 2: Determining whether Theorem 1 implies the existence of a unique solution or not

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.dxdt+cos x=sin t, x(π)=0

Step by Step Solution

Step 1: Finding the partial derivative of the given relation concerning y.

Step 2: Determining whether Theorem 1 implies the existence of a unique solution or not

Most popular questions for Math Textbooks

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In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dx}{dt} + \cos x = \sin t, x (π) = 0$