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Q12E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 186

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $y'' + y' + ty = e^{t} + 7$

No, the method of undetermined coefficients together with superposition can’t be applied.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the method of undetermined coefficients.

Given equation,

$y'' + y' + ty = e^{t} + 7$

Here, the given differential equation is non-homogeneous.

According to the method of undetermined coefficients, the method of undetermined coefficients applies only to non-homogeneities that are polynomials, exponentials, sine, cosine, or products of these functions.

And the left-hand side consists of the differential equation with constants coefficients.

Step 2: Conclusion.

The L.H.S. of the equation has a non-constant coefficient, $t$ .

Thus, the method of undetermined coefficients cannot be applied.

Most popular questions for Math Textbooks

Using the mass-spring analogy, predict the behavior as $t \to \infty$ of the solution to the given initial value problem. Then confirm your prediction by actually solving the problem.

$\begin{array}{l} (a) . y'' + 16 y = 0; y (0) = 2, y' (0) = 0 \\ (b) . y'' + 100 y' + y = 0; y (0) = 1, y' (0) = 0 \\ (c) . y'' - 6 y' + 8 y = 0; y (0) = 1, y' (0) = 0 \\ (d) . y'' + 2 y' - 3 y = 0; y (0) = - 2, y' (0) = 0 \\ (e) . y'' - y' - 6 y = 0; y (0) = 1, y' (0) = 1 \end{array}$

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation. $y'' + 2 y' - y = t^{- 1} e^{t}$

Find a general solution $y'' + 4 y' + 8 y = 0$

Find a general solution to the differential equation. $y'' (θ) + 2 y' (θ) + 2 y (θ) = e^{- θ} cosθ$

Find a particular solution to the differential equation. $4 y'' + 11 y' - 3 y = - 2 {te}^{- 3 t}$

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