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### 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"(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.

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:

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.