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Q16E

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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 453
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

In Problems 15-17,solve the given initial value problem x2y"+5xy'+4y=0.

y(1) =3 and y'(1) = 7

The solution of the given initial value problem is y=3x-2+13x-2 ln x.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Define Cauchy-Euler equations:

In mathematics, a Cauchy problem is one in which the solution to a partial differential equation must satisfy specific constraints specified on a hypersurface in the domain.

An initial value problem or a boundary value problem is an example of the Cauchy problem.

The equation will be in the form of, ax2y"+bxy'+cy=0.

Find the general solution:

The given equation is,

x2y"+5xy'+4y=0

Let L be the differential operator defined by the left-hand side of equation, that is,

L [x] (t) = x2y"+5xy'+4y

And let.

w (r,t) = xr

Substituting the w(r,t) in place of x(t), you get

L [w] (t)=x2 (xr)"+5x (xr)'+4 (xr)

=x2 (r (r-1)) xr-2 +5x (r) xr-1 +4xr

= (r2-r) xr +5rxr +4xr

=(r2+4r+4) xr

Solving the indicial equation,

r2+4r+4 =0

(r+2)2 =0

There are repeated roots at r= -2

Thus there are two linearly independent solutions given by

y1=c1x-2 and y2 = c2x-2 lnx

The general solution for the equation will be

y=c1x-2 +c2x-2 lnx

Determine the initial value:

For the given initial conditions,

y(1)=3 and y'(1)=7

y(x)=c1 x-2+c2 x-2 lnx

y(1) = c1

c1=3

y'(x)=c1 (-2) x-3 +c2 [(-2) x-3 lnx+x-2 (1/x)]

y'(1) = -2c1+c2

-2c1+c2=7

Solving the two simultaneous equations (1) and (2), you get the values of two constants c1 and c2 as,

c1 = 3 and c2 = 13

Thus the solution of the given initial value problem is y=13 x-2+13x-2 ln x.

Most popular questions for Math Textbooks

In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.

d2y/dx2=5/x dy/dx-13/x2 y

The equation

(1-x2)y"-2xy'+n(n+1)y=0

where n is an unspecified parameter is called Legendre’s equation. This equation appears in applications of differential equations to engineering systems in spherical coordinates.

(a) Find a power series expansion about x=0 for a solution to Legendre’s equation.

(b) Show that for a non negative integer there exists an nth degree polynomial that is a solution to Legendre’s equation. These polynomials upto a constant multiples are called Legendre polynomials.

(c) Determine the first three Legendre polynomials (upto a constant multiple).

In problems 1-6, determine the convergence set of the given power series.

∑n=0∞(n+2)!n!(x+2)n

For Duffing's equation given in Problem 13, the behaviour of the solutions changes as r changes sign. When r>0 , the restoring force ky+ry3 becomes stronger than for the linear spring(r=0). Such a spring is called hard. When r<0, the restoring force becomes weaker than the linear spring and the spring is called soft. Pendulums act like soft springs.

(a) Redo Problem 13 with r=-1. Notice that for the initial conditions ,y(0)=0,y'(0)=1 the soft and hard springs appear to respond in the same way for small.

(b) Keeping k=A=1 and ,ω=10 change the initial conditions to y(0)=1and y'(0)=0. Now redo Problem 13 with r=±1.

(c) Based on the results of part (b), is there a difference between the behavior of soft and hard springs for small? Describe.

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four non-zero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

w'+xw=ex
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