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Home Differential Equations Second Order Homogeneous Linear Equations Solving a Second Order Homogeneous Differential Equation with Constant Coefficients
 
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Solving a Second Order Homogeneous Differential Equation with Constant Coefficients

(1)

ay" + by' + cy = 0, a ≠ 0.

Step 1

Form the characteristic polynomial

az2 + bz + c.

Find its roots by using the quadratic equation or by factoring.

Step 2

The general solution is described by three cases.

Case 1: Two distinct real roots:

z = r, z = s.

y = Aert + Best.

Case 2: One real root:

z = r.

y = Aert + Btert.

Case 3: Two complex conjugate roots:

z = α + iβ.

y = eαt[A cos (βt) + B sin (βt)].

Step 3

If initial values for y and y' are given, solve for A and B, and substitute to obtain the particular solution. The two initial values will specify the position and velocity at one time:

y = y0 and y'= v0 axt = t0.

Discussion

The general solution in Case 3 is sometimes written in the same form as Case 1 by using complex exponents,

y = Cert + Dest,

where

r = α + iβ,

s = α - iβ,

C = ½(A- iB),

D = ½(A + iB).

To show that the two forms of the solution are really the same, use the complex exponent formula

eα+iβ = eα(cos β + i sin β)

from the preceding section.
Home Differential Equations Second Order Homogeneous Linear Equations Solving a Second Order Homogeneous Differential Equation with Constant Coefficients

Last Update: 2006-11-16