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[4. Applications.]
Linear differential equations.
A second order linear differential equation with constant coefficients is an equation of the type:
where a , b and c are real numbers.
Finding functions f(x) solutions of this kind of equations is simple. Consider the
second order equation a·k² + b·k + c = 0,
called the characteristic equation, and let k_{1} and k_{2}
be its solution. There are three different cases:

k_{1} and k_{2} are different real numbers. Two particular solutions
of equation (D.1) are f_{1}(x) = e^{ x k1}
and f_{2}(x) = e^{ x k2} .

k_{1} and k_{2} are equal real numbers. Two particular solutions
of equation (D.1) are are f_{1}(x) = e^{ x k1}
and f_{2}(x) = x·e^{ x k1} .

k_{1} and k_{2} are different complex numbers:
k_{1} = p + iq , k_{2} = p  iq.
Two particular solutions of equation (D.1) are
f_{1}(x) = e^{ p x} cos(qx) and
f_{2}(x) = e^{ p x} sin(qx) .
The general solution of equation (D.1) is a linear combination of two particular solutions:
f(x) = A·f_{1}(x) + B·f_{2}(x) ,
where A and B are real numbers.
