Q30E Superposition Principle. Let聽y1... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q30E (page 200)

Superposition Principle. Let $y_{1}$ be a solution to $y'' (t) + p (t) y' (t) + q (t) y (t) = g_{1} (t)$ on the interval Iand let $y_{2}$ be a solution to $y'' (t) + p (t) y' (t) + q (t) y (t) = g_{2} (t)$ on the same interval. Show that for any constants $k_{1}$ and $k_{2}$ , the function $k_{1} y_{1} + k_{2} y_{2}$ is a solution on Ito $y'' (t) + p (t) y' (t) + q (t) y (t) = k_{1} g_{1} (t) + k_{2} g_{2} (t)$ .

Short Answer

Expert verified

$k_{1} y_{1} + k_{2} y_{2}$ is the solution to the $y'' (t) + p (t) y' (t) + q (t) y (t) = k_{1} g_{1} (t) + k_{2} g_{2} (t)$ .

Step by step solution

Check whether or be the solution

If $y_{1}$ be the solution of the differential equation $y'' (t) + p (t) y' (t) + q (t) y (t) = g_{1} (t)$

Then $y_{1}^{} {'' (t) + p (t) y}_{1}^{}' (t) + q (t) y_{1} (t) = g_{1} (t)$ and if $y_{2}$ be the solution of the differential equation;

$y'' (t) + p (t) y' (t) + q (t) y (t) = g_{2} (t)$

Then $y_{2}^{} {'' (t) + p (t) y}_{2}^{}' (t) + q (t) y_{2} (t) = g_{2} (t)$

So, let's take $y = {ky}_{1} + {ky}_{2}$ then $y' = k_{1} y_{1}^{}' + k_{2} y_{2}^{}' y'' = k_{1} y_{1}^{}'' + k_{2} y_{2}^{}''$

Substitute the values

Substitute these equations in the differential equation:

$y'' (t) + p (t) y' (t) + q (t) y (t) = k_{1} y_{1}'' + k_{2} y_{2}'' + p (t) (k_{1} y_{1}^{}' + k_{2} y_{2}^{}') + q (t) ({ky}_{1} + {ky}_{2}) = k_{1} (y_{1}'' (t) + p (t) y_{1}' (t) + q (t) y_{1} (t)) + k_{2} (y_{2}'' (t) + p (t) y_{2}' (t) + q (t) y_{2} (t))$

Then from (1) and (2)

$y'' (t) + p (t) y' (t) + q (t) y (t) = k_{1} g_{1} (t) + k_{2} g_{2} (t)$

Therefore $k_{1} y_{1} + k_{2} y_{2}$ is the solutions to the $y'' (t) + p (t) y' (t) + q (t) y (t) = k_{1} g_{1} (t) + k_{2} g_{2} (t)$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Check whether or be the solution

Substitute the values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Mechanics Maths

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

91影视

Superposition Principle. Lety1be a solution to y''(t)+p(t)y'(t)+q(t)y(t)=g1(t) on the interval Iand lety2be a solution to y''(t)+p(t)y'(t)+q(t)y(t)=g2(t) on the same interval. Show that for any constantsk1and k2, the functionk1y1+k2y2is a solution on Ito y''(t)+p(t)y'(t)+q(t)y(t)=k1g1(t)+k2g2(t).

Short Answer

Step by step solution

Check whether or be the solution

Substitute the values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Mechanics Maths

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.