Linear Systems of Differential Equations with Complex Roots

Linear systems of differential equations can also feature characteristic equations with complex roots that lead to complex conjugate solutions. Consider the system:

dx/dt = 3x + 2y

dy/dt = -5x + y

 

Much of the procedure for solution of this system is the same as for the earlier examples. Thus, assume a solution of the form:

 

x = A exp(lt) and y = B exp(lt)

Now, substitute the preceding into the system to get:

 

l A exp(lt)   =  3 A exp(lt)    +  2B exp(lt)

lB exp(lt)   =    -5A exp(lt)     +  B exp(lt)

This leads to:
 

 

3 A exp(lt) - l A exp(lt)     + 2 B exp(lt) =  0

 

-5 A exp(lt)   + B exp(lt)   -lB exp(lt)   = 0

And the algebraic system:

 

(3 - l) A  + 2B = 0

-5A  +    (1 - l) B = 0

 

And then to the form: (A - l) D = 0 =
 

(3 - l………2)
(-5………1 - l )

And then expand to obtain the characteristic equation:

l² – 4l  + 13 = 0

Using the quadratic formula one can determine the roots of this system (with b = -4, a =1, and c = 13) are 2 +  3i.   Now, set l = 2 + 3i in the algebraic system to get:

(1 – 3i) A  + 2B = 0

 

-5A  +    -(1 – 3i) B = 0

A simple, nontrivial solution of this system (using the method to solve for A, B we employed before) yields: A = 2 and B = -1 + 3i. Using these one obtains the complex solutions:

 

x = 2 exp (2 + 3i)t

y =  (-1 + 3i) exp (2 + 3i)t

 

 

Which can alternatively be written (using Euler’s formula) :
 

x = exp (2t) [(2 cost 3t) + i(2 sin 3t)]

y = exp(2t)[(-cos 3t – 3 sin 3t) + i (3 cos 3t – sin 3t)]

Since both the real and imaginary parts of this solution are themselves solutions of the system, we also obtain the two real solutions:

x = 2 exp t cos 3t

y =  - exp 2t (cos 3t + 3 sin 3t)

and also:

x = 2 exp t sin 3t

y =  -exp 2t (3cos 3t -  sin 3t)

Since the preceding solutions are linearly independent we can write the general solution of the system as:

x = 2exp (2t) (c1 cost 3t + c2 sin 3t)

y = exp(2t)[c1(-cos 3t – 3 sin 3t) + c2 (3 cos 3t – sin 3t)]

 

Problems for Math Mavens:
 

1)   The other root of the characteristic equation was: 2 – 3i. Set l = 2 – 3i and find out what the general solution is to the system illustrated in the post.
 

2)     Find the complex and general solutions of the system:
 

dx/dt = 5x – 2y

 

dy/dt = 4x - y