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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 t2x"-12x=0. x(1)=3 and x'(1)=5.

The solution of the given initial value problem is x=2t4+t-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 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,

t2x"-12x=0

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

L [x] (t) = t2x"-12x

And let.

w (r,t) = tr

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

L [w] (t)=t2 (tr)"-12(tr)

=t2 (r (r-1)) tr-2-12(tr)

= (r2-r)tr- 12tr

=(r2-r-12)tr

Solving the indicial equation,

r2-r-12=0

(r-4)(r+3)=0

r=4, -3

There are two roots of the above indicial equation.

r1=4 and r2=-3

We can write two linearly independent real-valued solutions as,

x1=t4 and x2=t-3

The general solution for the equation will be,

x=c1t4+c2t-3

## Determine the initial value:

For the given initial conditions.

x(1) = 3 and x'(1)=5

x(t)= c1t4+c2t-3

x(1) = c1+c2

Here,

c1+c2 =3

Now,

x'(t)= 4c1t3-4c2t-4

x'(1)= 4c1-3c2

=5

Solving the two simultaneous equations, you get the values of two constants c1 and c2 as,

c1=2 and c2=1

Thus, the solution of the given initial value problem is,

x=2t4+t-3 ### Want to see more solutions like these? 