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Review of First and Second Order Linear ODEs
1) Consider the linear first order ODE with arbitrary coefficients:
y ′ + a(x )y = b(x )
Its complete solution (homogeneous and particular) is simply
− a dx 
y =e ∫
C 1 +

∫ be
+ ∫ a dx

dx 

2) Consider the second order linear ODE with constant coefficients:
y ′′ + ay ′ + by = 0
Assuming a general solution of the form y = exp(λx ) , the resulting characteristic or auxiliary
equation for the argument λ is the quadratic polynomial
λ 2 + aλ + b = 0
with roots
λ1 = 21 (−a + a 2 − 4b )
λ2 = 21 (−a − a 2 − 4b )
Three general cases can occur and these are summarized below.
Case
Roots
General solution
I: two distinct real
λ1 , λ2 given above
y = C 1e λ1x + C 2e λ2x
II: double real
λ1,2 = − 21 a , a = ±2 b
y = (C 1 + C 2x )e −ax / 2
III: complex conjugate
λ1,2 = − 21 a ± i ω , ω =
1
2
4b − a 2
y = (C 1 cos ωx + C 2 sin ωx )e −ax / 2