Differential Equations
Cauchy – Euler Differential Equation and Special 2nd Order DE
Cauchy Euler Differential Equation
Form: anxny(n) + an-1xn-1y(n-1) + … + a2x2y´´+a1xy´+ a0y = 0
dk y
Auxillary equation: ak x k dxk = ak m(m-1)(m-2)…(m-k+1)
Solution:
Case 1 : real and distinct roots
y = C1xm1 + C2xm2
Case 2: repeated real roots
y = C1xm + C2xmlnx
Case 3: conjuate complex roots
y = xa( C1cos(blnx) + C2sin(blnx))
Note: If the equation is non-homogeneous, g(x) ≠ 0 , apply the variation of parameters.
Special 2nd Order Differential Equation
d
dy
Case 1: dx { dx + yP(x) } = Q(x)
Solution:
dy
Let u = dx + yP(x)
Case 2: A second-ordwer differential equation with the variable y missing
d2 y dy
F ( dx2 , dx , x )
Solution:
dy
Let u = dx and
du
d2 y
=
dx dx2
Case 3: A second order differential equation with the variable x missing
d2 y dy
F ( dx2 , dx , y )
Solution:
dy
Let u = dx and
1
d2 y
dx2
du dy
du
= dx ∙ dy = u dy