Second Order Differential Equations - Generalities

Generalities

The general form of the second order linear differential equation is as follows

d2y / dx2 + P(x) dy / dx + Q(x) y = R(x)


If R(x) is not equal to zero, the above equation is said to be inhomogeneous.

If R(x) = 0, the above equation becomes

d2y / dx2 + P(x) dy / dx + Q(x) y = 0


and is called second order linear homogeneous differential equation.

Theorem

If y1(x) and y2(x) are two linearly independent solutions of the homogeneous differential equation d2y / dx2 + P(x) dy / dx + Q(x) y = 0, then the general solution of the above equation may be written as

y(x) = A y1(x) + B y2(x)


where A and B are constants.

NOTE: Functions y1(x) and y2(x) are linearly independent if one is not a multiple of the other.

Second Order Differential Equations With Constant Coefficients

Homogeneous second order differential equations with constant coefficients have the form

d2y / dx2 + b dy / dx + c y = 0


where b and c are constants.

Because of the presence of the first and second derivatives in the above equation, solutions of the form y = ekx are appropriate for the above equation.

If y = ekx, then dy / dx = k ekx and d2y / dx2 = k2 ekx.

Substitute y, dy/dx and d2y / dx2 into the differential equation to obtain

k2 ekx + b k ekx + c ekx = 0

Factor ekx out

ekx (k2 + b k + c ) = 0

and since ekx cannot be zero leads to

k2 + b k + c = 0


The above equation in k is called the auxiliary equation for the given homogeneous equation. The solutions k1 and k2 of the auxiliary equation are given by

k1 = [ - b + sqrt(D) ] / 2 and k1 = [ - b - sqrt(D) ] / 2

where D = b2 - 4c.

Since D may be negative, positive or equal to zero, solutions k1 and k2 may real and distinct when D > 0, real and equal when D = 0 and complex conjugate when D < 0. All these cases will be discussed in the following pages:

Solve Second Order Differential Equations - part 1

Solve Second Order Differential Equations - part 2

Solve Second Order Differential Equations - part 3



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Updated: 2 April 2013

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