Solving ay'' + by' + cy = 0, where a,b,c are constants

Write down the characteristic equation, which is a quadratic,

ar^2 + br + c = 0

Solve the characteristic equation, giving two roots, r1 and r2 using the quadratic formula

r1 = (−b/a + sqrt(b^2−4ac) ) / (2a)
r2 = (−b/a - sqrt(b^2−4ac) ) / (2a)

If r₁,r₂ are distinct real numbers, then the general solution is

y = c₁e^r1^x + c₂e^r2^x  

If r1=r2 then we will denote the root r, and the general solution is

y = c₁e^rx + c₂xe^rx  

If r₁,r₂ are complex numbers, then the general solution is

y = e^αx ( c₁ cos(βx) + c₂ sin(βx) ), where  α and β are the real and imaginary components, respectively of r₁.

Why does this work?

If r₁,r₂ are distinct numbers, real or not, then the general solution is still

y = c₁e^r1^x + c₂e^r2^x  

In the case where r₁,r₂ are complex, we can consider the real and imaginary components separately, since r1=α+βi and r2=α−βi.

y = c₁e^(α+βi)x + c₂e^(α−βi)x  
 =  e^(α) ( c₁e^βi + c₂e^−βix )
 =  e^(α) (  c₁ cos(βx) + c₁ i sin(βx) +  c₂ cos(−βx) + c₂ i sin(−βx))
 =  e^(α) ( (c₁+c₂)cos(βx) + (c₁−c₂) i sin(βx)))

Now, let d1=c₁+c₂, and let d2=c₁i−c₂i.  This gives us

y = e^αx ( d₁ cos(βx) + d₂ sin(βx) )

Now, of course the selection of a letter to represent the constant is arbitrary, so this is the solution that was presented above.

Example

y'' + 2y' + 5y = 0

r^2 + 2r + 5 = 0

r = -1 ± 2i

α = -1 and β = 2, so

y = e^-x ( c₁ cos(2x) + c₂ sin(2x) )

checking that it is in fact a solution...

y' = e^-x (2 c₂ cos(2x) - 2 c₁ sin(2x)) - (c₁ cos(2x) + c₂ sin(2x))
  = e^-x ((-c₁+2c₂)cos(2x) + (-2c₁-c₂)sin(2x))

y'' = e^-x (-2 (2 c₂ cos(2x) - 2 c₁ sin(2x))) + e^-x (-4 c₁ cos(2x) - 4 c₂ sin(2x)) + e^-x (c₁ cos(2x) + c₂ sin(2x))
   =   e^-x ((-3c₁ - 4c₂) cos(2x) + (4c₁ - 3c₂)sin(2x)) 

Putting it together, and substituting into the original quadratic,

5 e^-x (c₁ cos(2x) + c₂ sin(2x)) +
2 e^-x ((-c₁+2c₂)cos(2x) + (- 2c₁-c₂)sin(2x)) +
  e^-x ((-3c₁ - 4c₂) cos(2x) + (4c₁ - 3c₂)sin(2x))

= 0

Internet References

SOS Math: Homogeneous Linear Equations with Constant Coefficients 

SOS Math: Homogeneous Linear Equations of the form a(x)y'' + b(x)y' + c(x)y = 0

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