second order differential equation

數學神童

using the substitution x=e^z,show that the differential equation
x^2(d^2y/dx^2)+px(dy/dx)+qy=0 where p and q are constants,can be transformed into the differential equation
(d^2y/dz^2)+r(dy/dz)+sy=0 where r and s are constants to be determined in terms of p and q.

iamverynoob

x=e^z
dx/dz=e^z   -> dz/dx = 1/e^z = 1/x

dy/dx = dy/dz * dz/dx = dy/dz * 1/x
d^2y/dx^2 = d(dy/dx)/dx = d(dy/dz * 1/x)/dz * dz/dx = (d^2y/dz^2 * 1/x) * 1/x ＝ d^2y/dz^2 * 1/x^2

所以
x^2(d^2y/dx^2)+px(dy/dx)+qy=0
变成
x^2(d^2y/dz^2 * 1/x^2) + px(dy/dz * 1/x) + qy = d^2y/dz^2 + pdy/dz + qy = d^2y/dz^2 + rdy/dz + sy = 0