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Q23E

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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 337
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

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

use the annihilator method to determinethe form of a particular solution for the given equation y''-5y'+6y=e3x-x2

y''-5y'+6y=e3x-x2

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Solve the homogeneous of the given equation

The homogeneous of the given equation is

D2-5D+6[y]=(D-2)(D-3)[y]=0

The solution of the homogeneous is

yh(x)=c1e2x+c2e3x (1)

Let g(x)=e3x

Then

(D-3)[g]=0

Let h(x)=x2

Then

D3[h]=0

Hence

D3(D-3)[g-h]=0

Then, every solution to the given nonhomogeneous equation also satisfies

. D3(D-3)(D-2)(D-3)[y]=D3(D-2)(D-3)2[y]=0

Then

y(x)=c1e2x+c2e3x+c3xe3x+c4+c5x+c6x2 (2)

is the general solution to this homogeneous equation

We know u(x)=uh+up

Comparing (1) & (2)

yp(x)=c3xe3x+c4+c5x+c6x2

Most popular questions for Math Textbooks

Given that {x,x-1,x4} is a fundamental solution set for the homogeneous equation corresponding to the equationx3y'''-x2y''-4xy'+4y=g(x), x>0, determine a formula involving integrals for a particular solution.

Higher-Order Cauchy–Euler Equations. A differential equation that can be expressed in the form

anxny(n)(x)+an−1xn−1y(n−1)(x)+...+a0y(x)=0

where an,an−1,....,a0 are constants, is called a homogeneous Cauchy–Euler equation. (The second-order case is discussed in Section 4.7.) Use the substitution y=xy to help determine a fundamental solution set for the following Cauchy–Euler equations:

(a) x3y'''+x2y''−2xy'+2y=0,x>0

(b) x4y(4)+6x3y'''+2x2y''−4xy+4y=0,x>0

(c) x3y'''−2x2y''+13xy'−13y=0,x>0

[Hint: xα+βi=e(α+βi)lnx=xa{cos(βlnx)+isin(βlnx)}]

Find a general solution to y’’’ - 3y’ - y = 0 by using Newton’s method or some other numerical procedure to approximate the roots of the auxiliary equation.

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.

z'''+3z''-4z=e2x

Deflection of a Beam Under Axial Force. A uniform beam under a load and subject to a constant axial force is governed by the differential equation

y(4)(x)-k2y''(x)=q(x), 0<x<L,

where is the deflection of the beam, L is the length of the beam, k2 is proportional to the axial force, and q(x) is proportional to the load (see Figure 6.2).

(a) Show that a general solution can be written in the form

y(x)=C1+C2x+C3ekx+C4e-kx+1k2∫q(x)xdx-xk2∫q(x)dx+ekx2k3∫q(x)e-kxdx-e-kx2k3∫q(x)ekxdx

(b) Show that the general solution in part (a) can be rewritten in the form

y(x)=c1+c2x+c3ekx+c4e-kx+∫0xq(s)G(s,x)ds,

where

G(s,x):=s-xk2-sinh[k(s-x)]k3.

(c) Let q(x)=1 First compute the general solution using the formula in part (a) and then using the formula in part (b). Compare these two general solutions with the general solution

y(x)=B1+B2x+B3ekx+B4e-kx-12k2x2,

which one would obtain using the method of undetermined coefficients.

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