Second Order Differential Equations - Generalities
Generalities
The general form of the second order linear differential equation is as follows
If R(x) is not equal to zero, the above equation is said to be inhomogeneous. If R(x) = 0, the above equation becomes 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
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
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 The above equation in k is called the auxiliary equation for the given homogeneous equation. The solutions k1 and k2 of the auxiliary equation, which is a quadratic equation in k, are given by k1 = [ - b + √D ] / 2 and k1 = [ - b - √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 |
