Second Order Unforced Equations Complementary Solutions

1 Second Order Unforced Equations Complementary Solutions 

The second order linear equation is of great importance and arises frequently in engineering. We shall reserve treatment of the case of nonconstant coefficients that is. 

for constant coefficients, we shall assume there exists complementary solutions of the form 

There are three ways to satisfy this equation, two of them are trivial (remember, zero is always a solution to homogeneous equations, a trivial result). We thereby deduce that the root(s) must be satisfied by 

It is easy to see that each of these, taken one at a time, satisfies the original equation, so that the general solution is the sum of the two. But are the solutions linearly independent To answer this, we need to know the nature of the two roots. Are they real or complex Are they unequal or equal We consider these possibilities by considering a series of examples. 

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