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 Skip Navigation LinksMath Help > Calculus > Differential Equations > Homogeneous Linear DE

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 = c1er1x + c2er2x  

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

y = c1erx + c2xerx  

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

y = eαx ( 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 = c1er1x + c2er2x  

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) ( c1eβi + 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 = eαx ( 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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