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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 450
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 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.

x2y"(x)+6xy'(x)+6y(x)=0

The general solution to the given equation x2y"(x)+6xy'(x)+6y(x)=0 is y=c1x-2+c2x-3.

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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 both examples of Cauchy problems. The equation will be in the form of ax2y"+bxy'+cy=0.

Find the general solution:

Given,

x2y"(x)+6xy'(x)+6y(x)=0

Let L be the differential operator defined by the left hand side of the equation.

L[y](x)=x2y"+6xy'+6y

w(r,x)=xr

By substituting you get,

L[y](x)=x2(xr)"+6x(xr)'+6xr

=x2(r(r-1)) xr-2+6x(r)xr-1+6xr

=(r2-r) xr +6rxr +6xr

=(r2+5r+6) xr

Solving the indicial equation.

r2+5r+6 =0

(r+3) (r+2)=0

The two distinct roots are,

r1=-2

r2=-3

There are two linearly independent solutions.

y1=c1x-2

y2=c2x-3

The general solution is y=c1x-2+c2x-3.

Most popular questions for Math Textbooks

(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).

(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.

The solution to the initial value problem

xy''(x)+2y'(x)+xy(x)=0y(0)=1,   y'(0)=0

has derivatives of all orders at x=0 (although this is far from obvious). Use L'Hôpital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

y'-exy=0; y(0)=1

In Problems 11 and 12, use a substitution of the form to find a general solution to the given equation for x>c.

4(x+2)2y"+5y=0

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

y"-xy'+2y=cosx
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