Q5E - 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.

d²y/dx²=5/x dy/dx-13/x² y

Step-by-Step Solution

Answer

The general solution for the given equation is y=c₁x³  cos (2 lnx)+c₂x³ sin(2 lnx).

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:

The given equation is

d²y/dx²=5/x dy/dx-13/x² y

This can be re-written in the required form as

x²y"-5xy'+13y=0

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

L[y](x) = x²y"-5xy'+13y

And let's set,

w(r,x)=x^r

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

L[y](x) = x²(x^r )"-5x(x^r)'+13(x^r)

=x² (r(r-1)) x^r-2 -5x(r) x^r-1 +13x^r

=(r²-r) x^r -5rx^r+13x^r

=(r²-6r+13) x^r

Solving the indicial equation,

r²-6r+13=0

r=[2± √(36-4×13)]/2

=(6± 4i)/2

=3±2i

There are two complex conjugates,

r₁=3+2i   and   r₂=3-2i 

Thus there are two linearly independent solutions given by,

x³⁺²ⁱ=x³ cos(2 lnx)+ix³ sin (2 lnx)

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

y₁=x³ cos(2 lnx)  and  y₂=x³ sin(2 lnx)

Therefore, the general solution for the equation will be,

y=c₁ x³ cos(2 lnx)+c₂ x³ sin(2 lnx)