Question

What is a second solution to the Differential Equation `x^2y'' -3xy'+5y=0`?

We are given that the answer does not contain cosine functions.

Answer 1

See below.

Explanation:

Assuming that the differential equation reads

`x^2y''-3x y'+5y=0`

The differential equation

`x^2y''-3x y'+5y=0` is a linear homogeneous differential equation.

Proposing

`y = c_0 x^alpha` and substituting

`(alpha(alpha-4)+5)c_0x^alpha = 0` and solving for `alpha`

`alpha(alpha-4)+5=(alpha-2+i)(alpha-2-i)`

then

`y = x^2(c_1 x^i + c_2 x^-i)`

but `x = e^(log_e x)` and `x^(pm i)=e^(pm i log_e x)`

now according to

`e^(i z) = cos z + i sin z` we have the general solution.

`y = x^2(C_1 sin (log_e x) + C_2 cos(log_e x ))`

Answer 2

Second solution is:

`y=Bx^2sin(lnx)`

Explanation:

If we assume the a corrected equation:

`x^2y'' -3xy'+5y=0` ..... [A]

This is a Euler-Cauchy Equation which is typically solved via a change of variable. Consider the substitution:

`x = e^t => xe^(-t)=1`

Then we have,

`dy/dx = e^(-t)dy/dt`, and, `(d^2y)/(dx^2)=((d^2y)/(dt^2)-dy/dt)e^(-2t)`

Substituting into the initial DE [A] we get:

`x^2((d^2y)/(dt^2)-dy/dt)e^(-2t) -3xe^(-t)dy/dt+5y=0`

`:. ((d^2y)/(dt^2)-dy/dt) -3dy/dt-+5y=0`

`:. (d^2y)/(dt^2)-4dy/dt+5y=0` ..... [B]

This is now a second order linear homogeneous Differentiation Equation. The standard approach is to look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.

`m^2-4m+5 = 0`

We can solve this quadratic equation, and we get two complex conjugate roots:

`m=2+-i`

Thus the Homogeneous equation [B]:

`(d^2y)/(dt^2)-4dy/dt+5y=0`

has the solution:

`y=e^(2t)(Acost+Bsint)`

Now we initially used a change of variable:

`x = e^t => t=lnx`

So restoring this change of variable we get:

`y=(x)^2(Acos(lnx)+Bsin(lnx))`

`:. y=x^2(Acos(lnx)+Bsin(lnx))`

`:. y=Ax^2cos(lnx)+Bx^2sin(lnx)`

Which is the General Solution.

From the given answer we can clearly infer that for the first part of the solution `A=0`, but we cannot make any further assumptions about the constant `B` in the second solution