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Expert-verified Found in: Page 450 ### 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 # In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0. x2y"+2xy'-3y=0

The general solution for the given equation is y=c1x-1/2+√13/2 +c2x-1/2-√13/2 .

See the step by step solution

## 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:

The given equation is,

x2y"+2xy'-3y=0

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

L[y](x)=x2y"+2xy'-3y

Let's see,

w(r,x)=xr

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

L[y](x)=x2(xr )"+2x(xr)'-3(xr)

=x2 (r(r-1)) xr-2+2x (r) xr-1-3 (xr)

=(r2-r) xr+2rxr-3xr

=(r2+r-3) xr

Solving the indicial equation

r2+r-3=0

r= [-1±√(1+12)]/2

r= -1/2±√13/2

There are two distinct roots,

r1= -1/2+√13/2 and r2= -1/2-√13/2

Thus there are two linearly independent solutions given by,

y1=c1x-1/2+√13/2 and y2=c2x-1/2-√13/2

Hence, the general solution for the given equation will be,

y=c1x-1/2+√13/2 +c2x-1/2-√13/2

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