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Homogeneous linear differential equations with constant coefficients, Auxiliary equation, solutions


We shall here treat the problem of finding the general solution to the homogeneous linear differential equation with constant coefficients


ole.gif


Such an equation can be written in the operator form

 

ole1.gif


or, more simply,


            f(D)y = 0


where f(D) is a differential operator.



Def. Auxiliary equation. The auxiliary equation of equation 1) is the equation


            f(m) = a0 mn + a1 mn -1 + ...... + an-1 m + an = 0


The function f(m) may be obtained by simply substituting m for D in the differential operator

 

             ole2.gif


of equation 2).



Theorem. If m is any root of f(m) = 0 , then


            f(D)emx = 0,


which means that y = emx is a solution of equation 1).


Proof


The auxiliary equation has n roots. Let the n roots be m1, m2, ..... , mn. If these roots are all real and distinct, then it can be shown that the n solutions


             ole3.gif


are linearly independent. The general solution is then given by


             ole4.gif


where c1, c2, ..... , cn are arbitrary constants.


However some roots may not be real and distinct. Some roots may be multiple roots, complex roots or both. The set of all real, distinct roots make one contribution to the general solution. Each set of multiple roots makes its own separate contribution. The complex roots make their own contribution. The general solution consists of the sum of the contributions from these different sets.


Contribution from the set of all real, distinct roots. Suppose there are k real, distinct roots. Let the k roots be m1, m2, ..... , mk. Then the contribution to the general solution from these k real, distinct roots is


             ole5.gif


Example. Suppose there are three real roots 2, 5 and -3. Then the contribution to the general solution from these three real roots would be


            y1 = c1e2x + c2e5x + c3e-3x



Contribution from multiple (or repeated) roots. Suppose a root with a value m = b occurs with a multiplicity q (i.e. the auxiliary equation gives q equal roots with value m = b). Then the contribution to the general solution from this set of q equal roots is


             ole6.gif


Proof


Example. Suppose there are four equal roots 2, 2, 2, 2. Then the contribution to the general solution from these four equal roots would be


            y2 = c1e2x + c2xe2x + c3x2e2x + c4x3e2x




Contribution from complex roots. The contribution to the general solution from two conjugate complex roots a + bi and a - bi is

 

             ole7.gif


Repeated complex roots lead to solutions analogous to those for repeated real roots. For example, if the roots m = a ole8.gif bi occurred three times, the contribution would be


            (c1 + c2x + c3x2)eax cos bx + (c4 + c5x + c6x2)eax sin bx


Example. Suppose the auxiliary equation had the complex roots 2 + 3i and 2 - 3i. Then the contribution to the general solution from these two roots would be


            y3 = c1e2x cos 3x + c1e2x sin 3x

 


General solution. The general solution is given by


            y = yd + yr + yc


where yd is the contribution from all the distinct real roots, yr is the contribution from all the repeated roots, and yc is the contribution from all complex roots.




Summary of Procedure


1. Put differential equation in operator form

 

             ole9.gif


2. Form auxiliary equation by replacing the D’s in the operator form of the equation with m.


3. Find the roots of the auxiliary equation (using a numerical procedure if necessary).


4. Find the contributions to the general solution from the distinct, real roots, any repeated roots and any complex roots.


5. The general solution is given by


            y = yd + yr + yc


where yd is the contribution from all the distinct real roots, yr is the contribution from all the repeated roots, and yc is the contribution from all the complex roots.



Example. Solve the equation


            (D4 - 7D3 + 18D2 - 20D + 8)y = 0


The auxiliary equation is


            m4 - 7m3 + 18m2 - 20m + 8 = 0


which has the roots m = 1, 2, 2, 2. The general solution is


            y = c1ex + c2e2x + c3 xe2x + c3 x2e2x



Example. Solve the equation


            (D3 - 3D2 + 9D + 13)y = 0


The auxiliary equation is


            m3 - 3m2 + 9m + 13 = 0


which has the roots m = -1, 2 + 3i, 2 - 3i. The general solution is


            y = c1e-x + c2e2x cos 3x + c3e2x sin 3x




References

1. Earl D. Rainville. Elementary Differential Equations


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