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Expert-verified Found in: Page 453 ### 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 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.

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 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. ### Want to see more solutions like these? 