Chapter 4: Q8E (page 199)

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.
$(1 - t) y'' + ty' - 2 y = sint; y (0) = 1, y' (0) = 1$

Short Answer

Expert verified

The differential equation has a unique solution.

Step by step solution

Find the value of p(t),q(t),g(t)

The given differential equation is $(1 - t) y'' + ty' - 2 y = sint$ .

It can be written as $y'' + \frac{t}{(1 - t)} y' - \frac{2}{(1 - t)} y = \frac{sint}{(1 - t)}$

So, $p (t) = \frac{t}{(1 - t)}, q (t) = - \frac{2}{(1 - t)}, g (t) = \frac{sint}{(1 - t)}$

Check the result

From theorem (5) If p(t), q(t), and g(t) are continuous on an interval (a, b) that contains point t, then for any choice of the initial values $Y_{o} and Y_{1}$ , there exists a unique solution y(1) on the same interval (a, b) to the initial value problems.

Here p(t),q(t),g(t) is a continuous function in the interval but shows discontinuity at t=1.

So, theorem (5) applies except for point t=1.

Since the initial conditions are not the point (1,0) but are at (0,1).

Therefore the differential equation has a unique solution.s

This is the required result.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.
$(1 - t) y'' + ty' - 2 y = sint; y (0) = 1, y' (0) = 1$

Short Answer

Step by step solution

Find the value of p(t),q(t),g(t)

Check the result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Mechanics Maths

Statistics

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.(1-t)y''+ty'-2y=sint;y(0)=1,y'(0)=1

Short Answer

Step by step solution

Find the value of p(t),q(t),g(t)

Check the result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Mechanics Maths

Statistics

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.
$(1 - t) y'' + ty' - 2 y = sint; y (0) = 1, y' (0) = 1$