Chapter 1: Q24 E (page 14)

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dy}{dt} - ty = \sin^{2} t, y (π) = 5$

Short Answer

Expert verified

The hypotheses of Theorem 1 are satisfied.

It follows from the theorem that the given initial value problem has a unique solution.

Step by step solution

Finding the partial derivative of the given relation concerning y.

Here, $f (t, y) = \sin^{2} t + ty$ and $\frac{\partial f}{\partial y} = t$ .

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

Now from Step 1, we find that both of the functions $f (t, y)$ and $\frac{\partial f}{\partial y}$ are continuous in any rectangle containing the point $(π, 5)$ , 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 about $t = π$ of the form role="math" localid="1664002842140" $(π - δ, π + δ)$ , where is some positive number.

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

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In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dy}{dt} - ty = \sin^{2} t, y (π) = 5$

Short Answer

Step by step solution

Finding the partial derivative of the given relation concerning y.

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.dydt-ty=sin2t,y(π)=5

Short Answer

Step by step solution

Finding the partial derivative of the given relation concerning y.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Geometry

Decision Maths

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$\frac{dy}{dt} - ty = \sin^{2} t, y (π) = 5$