Short Answer

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dy}{dx} = y^{4} - x^{4}, y (0) = 7$

The hypotheses of Theorem 1 are satisfied.

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding the partial derivative of the given relation with respect to y

Here, $f (x, y) = y^{4} - x^{4}$ and $\frac{\partial f}{\partial y} = 4 y^{3}$ .

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 $f (x, y)$ and $\frac{\partial f}{\partial y}$ are continuous in any rectangle containing the point $(0, 7)$ , so the hypotheses of the Theorem are satisfied. It then follows from the theorem that the given initial value problem has a unique solution in an interval about $x = 0$ of the form $(0 - δ, 0 + δ)$ , 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.
$\frac{dy}{dx} = y^{4} - x^{4}, y (0) = 7$

Step by Step Solution

Step 1: Finding the partial derivative of the given relation with respect to 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.dydx=y4-x4, y(0)=7

Step by Step Solution

Step 1: Finding the partial derivative of the given relation with respect to y

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

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In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dy}{dx} = y^{4} - x^{4}, y (0) = 7$