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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 14

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

Question: In Problems 3–8, determine whether the given function is a solution to the given differential equation.
$y = 3 \sin 2 x + e^{- x}$ , $y'' + 4 y = 5 e^{- x}$

The given function is a solution to the given differential equation.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Differentiating the given equation w.r.t. (with respect to) x

Firstly, we will differentiate $y = 3 \sin 2 x + e^{- x}$ with respect to x,

$\frac{dy}{dx} = 6 \cos 2 x - e^{- x}$

Again, differentiating the given function with respect to x,

$\frac{d^{2} y}{{dx}^{2}} = - 12 \sin 2 x + e^{- x}$

Step 2: Simplification

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

$y'' + 4 y = - 12 \sin 2 x + e^{- x} + 4 (3 \sin 2 x + e^{- x}) y'' + 4 y = - 12 \sin 2 x + e^{- x} + 12 \sin 2 x + 4 e^{- x} y'' + 4 y = 5 e^{- x}$

which is the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, $y = 3 \sin 2 x + e^{- x}$ is a solution to the differential equation $y'' + 4 y = 5 e^{- x}$ .

Most popular questions for Math Textbooks

Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length l(t)of the wire varies with time in some predetermined fashion. If

U(t) is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem $l^{2} (t) θ'' (t) + 2 l (t) l' (t) θ' (t) + gl (t) \sin (θ (t)) = 0; θ (0) = θ_{o}, θ' (0) = θ_{1}$ where g is the acceleration due to gravity. Assume that $l (t) = l_{o} + l_{1} \cos (ωt - ϕ)$ where $l_{1}$ is much smaller than $l_{o}$ . (This might be a model for a person on a swing, where the pumping action changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take g = 1. Using the Runge– Kutta algorithm with h = 0.1, study the motion of the pendulum when $θ_{o} = 0.05, θ_{1} = 0, l_{o} = 1, l_{1} = 0.1, ω = 1, ϕ = 0.02$ . In particular, does the pendulum ever attain an angle greater in absolute value than the initial angle $θ_{o}$ ?

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

$\frac{dy}{dx} = y^{4} - x^{4}, y (0) = 7$

Stefan’s law of radiation states that the rate of change in the temperature of a body at T (t) kelvins in a medium at M (t) kelvins is proportional to $M^{4} - T^{4}$ . That is, $\frac{dT}{dt} = K [M {(t)}^{4} - T {(t)}^{4}]$ where K is a constant. Let $K = 2.9 \times 10^{- 10} {(\min)}^{- 1}$ and assume that the medium temperature is constant, M (t) = 293 kelvins. If T (0) = 360 kelvins, use Euler’s method with h = 3.0 min to approximate the temperature of the body after

(a) 30 minutes.

(b) 60 minutes.

Consider the question of Example 5 $y \frac{dy}{dx} - 4 x = 0$

Does Theorem 1 imply the existence of a unique solution to (13) that satisfies $y (x_{0}) = 0$ ?
Show that when $x_{0} \neq 0$ equation (13) can’t possibly have a solution in a neighbourhood of $x = x_{0}$ that satisfies $y (x_{0}) = 0$ .
Show two distinct solutions to (13) satisfying $y (0) = 0$ ( See Figure 1.4 on page 9).

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

$6 - t^{3} (\frac{d^{2} v}{d t^{2}}) + 3 v - l n t = \frac{d v}{d t}$

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