Q16E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 15-17,solve the given initial value problem x²y"+5xy'+4y=0.

y(1) =3 and y'(1) = 7

Step-by-Step Solution

Answer

The solution of the given initial value problem is y=3x^-2+13x^-2 ln x.

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) = x^r

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

L [w] (t)=x² (x^r)"+5x (x^r)'+4 (x^r)

=x² (r (r-1)) x^r-2 +5x (r) x^r-1 +4x^r

= (r²-r) x^r +5rx^r +4x^r 

=(r²+4r+4) x^r

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.