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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