Introduction:
Cauchy's homogenous linear differential equation d²y -2x-3y=0 dx can be solved using a substitution method.

Substitution methods:
There are various substitution methods available to solve the given differential equation. Let's discuss them one by one.

i) Substituting a bx=e²:
This substitution method is used when the differential equation is in the form of d²y/dx² + P(x)dy/dx + Q(x)y = 0, where P(x) and Q(x) are functions of x. Here, we substitute y = e^(bx) and solve for b. But in this case, the differential equation is not in the required form. So, this substitution method cannot be used.

ii) Substituting X:
This substitution method is used when the differential equation is of the form d²y/dx² + P(x)dy/dx + Q(x)y = 0, where P(x) and Q(x) are functions of x and P(x) and Q(x) have a common factor of degree 1. Here, we substitute y = Xv and solve for v. But in this case, the differential equation does not have a common factor of degree 1. So, this substitution method cannot be used.

iii) Substituting x = log z:
This substitution method is used when the differential equation is of the form d²y/dx² + P(x)dy/dx + Q(x)y = 0, where P(x) and Q(x) are functions of x and P(x) and Q(x) can be expressed as functions of log x. Here, we substitute x = e^z and y = u(z) and solve for u(z). But in this case, the differential equation cannot be expressed as a function of log x. So, this substitution method cannot be used.

iv) None of the above:
Since none of the above substitution methods can be used to solve the given differential equation, we need to use another method to solve it. One possible method is to use the power series method where we assume the solution to be a power series and determine the coefficients by substituting it in the differential equation.

Conclusion:
To solve the given differential equation, we cannot use the substitution methods i) a bx=e², ii) X, or iii) x = log z. Hence, we need to use another method like the power series method to solve it.