Reduction of Order

This technique is very important since it helps one to find a second solution independent from a known one. Therefore, according to the previous section, in order to find the general solution to y'' + p(x)y' + q(x)y = 0, we need only to find one (non-zero) solution, y₁.

Let y₁ be a non-zero solution of

Then, a second solution y₂ independent of y₁ can be found as

Easy calculations give

,

where C is an arbitrary non-zero constant. Since we are looking for a second solution one may take C = 1, to get

Remember that this formula saves time. But, if you forget it you will have to plug y₁v(x) into the equation to determine v(x) which may lead to mistakes !

The general solution is then given by

Example

Let u(w) be a utility function for wealth w. The function ρ(w) = -wu''(w)/u'(w) is known as the Arrow-Pratt measure of relative risk aversion. (If ρu(w) > ρv(w) for two utility functions u and v then u reflects a greater degree of risk-aversion than does v.)

What utility functions have a degree of risk-aversion that is independent of the level of wealth? That is, for what utility functions u do we have a = -wu''(w)/u'(w) for all w?

This is a second-order differential equation in which the term u(w) does not appear. Define z(w) = u'(w). Then we have a = -wz'(w)/z(w) or az(w) = -wz'(w), a separable equation that we can write as a·dw/w = -dz/z.

The solution is given by a·ln w = -ln z(w) + C, or z(w) = Cw^-a.

Now, z(w) = u'(w), so to get u we need to integrate:

u(w) =		Cln w + B, if a = 1
		Cw1-a/(1 - a) + B, if a ≠ 1

We conclude that a utility function with a constant degree of risk-aversion equal to a takes this form.