There's this problem that's asking me to use Euler's equidimensional equation, which is a second order differential equation. The problem tells me to substitute x = exp(z) to transform the equation into one with constant coefficients, and apply the technique to various examples.
Just to illustrate, Euler's equidimensional equation looks like this:
(x^2)*y'' + p*x*y' + q*y = 0, where p and q are constants. One of my tasks is to use the substitution x = exp(z) to solve the following problem:
(x^2)*y'' + 3*x*y' + 10*y = 0
Anyone have an idea how to use the substitution to my advantage? Just inserting exp(z) for x doesn't work. Should I be changing y' to be in terms of z?
Any help appreciated, thanks.
Put x = exp(z), then:
Originally Posted by primasapere
z = ln(x)
dz/dx = 1/x = exp(-z).
y' = dy/dx = dy/dz dz/dx= exp(-z) dy/dz
y'' = d^2y/dz^2 =d/dz[exp(-z) dy/dz)] dz/dx
....= [-exp(-z)dy/dz + exp(-z) d^2/dz^2] exp(-z)
....= exp(-2z) d^2y/dz^2 - exp(-2z) dy/dz.
Substituting these expressioons for x, y', and y'' into
x^2 y'' + 3 x y' + 10 = 0, gives:
d^2y/dz^2 -dy/dz + 3 dy/dz + 10 = 0,
d^2y/dz^2 + 2 dy/dz + 10 = 0.
Wow. That sort of approach was completely over my head. Thanks a lot!