Q2E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0. 

2x²y"(x)+13xy'(x)+15y(x)=0

Step-by-Step Solution

Answer

The general solution to the given equation 2x²y"(x)+13xy'(x)+15y(x)=0 is y=c₁x^-5/2+c₂x^-3.

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 both examples of Cauchy problems. The equation will be in the form of ax²y"+bxy'+cy=0.

Find the general solution:

As given,

2x²y"(x)+13xy'(x)+15y(x)=0

Let L be the differential operator defined by the left hand side of the equation.

L[y](x)=2x²y"(x)+13xy'(x)+15y(x) 

w(r,x)=x^r

By substituting you get,

L[w](x)=2x²(x^r)"+13x(x^r )'+15(x^r )

=2x² (r(r-1))x^r-2+13x(r)x^r-1+15x^r

=2(r²-r) x^r+13rx^r+15x^r

=(2r²+11r+15) x^r

Solving the indicial equation.

2r²+11r+15=0

(2r+5)(r+3)=0

The two distinct roots are,

r₁==5/2

r₂= -3

There are two linearly independent solutions.

y₁=c₁x^-5/2

y₂=c₂x^-3

The general solution is y=c₁x^-5/2+c₂x^-3.