Euler Equations - Series Solutions to Differential Equations

In this section we require to look for solutions to,

ax² y′′ + bxy′ + cy = 0

around x0  = 0. These kinds of differential equations are termed as Euler Equations.

Recall from the earlier section which a point is an ordinary point if the quotients the quotients,

(bx)/(ax²) = (b/ax)                               and c/(ax²)

Above equations have Taylor series around x₀ = 0. Though, due to the x in the denominator neither of these will contain a Taylor series around x₀ = 0 and so x₀ = 0 is a particular point.  Therefore, the method from the earlier section won't work as it needed an ordinary point.

Though, this is possible to find solutions to this differential equation that aren't series solutions. Here we start off through assuming that x>0 (the purpose for it will be apparent after we work the first illustration) and which all solutions are of the form,

y (x) = x^r    ............... (2)

 Here plug this in the differential equation to find,

ax²(r)(r-1) x^r-2 + bx (r) x^r-1 + cx^r = 0

ar(r-1) x^r + b(r) x^r + cx^r = 0

(ar(r-1) + b(r) + c)x^r = 0

Now, we assumed that x>0 and so this will only be zero if,

ar ( r -1) + b (r) +  c = 0   ...................       (3)

Therefore, solutions will be of the form (2) given r is a solution to (3). That equation is a quadratic in r and thus we will have three cases to look at: Distinct Roots, Real and Double Roots and Quadratic Roots.