Question

In Problems 15-17, solve the given initial value problem t²x"-12x=0. x(1)=3 and x'(1)=5.

Short answer

The solution of the given initial value problem is x=2t⁴+t^-3.

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) = t^r

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

L [w] (t)=t² (t^r)"-12(t^r)

=t²(r (r-1)) t^r-2-12(t^r)

= (r²-r)t^r- 12t^r

=(r²-r-12)t^r

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