Basics of Second Order, Ordinary Differential Equations

In a Nut Shell: A differential equation, y ’’ + p(x)y ’ + q(x)y = 0 is of the second

order since the second derivative of the dependent variable, y, appears in the d.e. This

d.e. appears frequently in mechanical systems. (i.e. vibration applications)

The principle of superposition applies to second order, linear, ordinary d.e.’s.

The sum of any two linearly independent solutions, y₁ and y₂ of the differential equation

y ’’ + p(x)y ’ + q(x)y = 0

y = c₁ y₁ + c₂ y₂ is again a solution of the differential equation

where c₁ and c₂ are constants.

y₁ and y₂ are linearly independent.

Note: The two functions, y₁ and y₂ , must be linearly independent.

Two functions , y₁ and y₂ , are linearly independent provided that neither is a

constant multiple of the other. The Wronskian, WR, is nonzero if the functions y₁

and y₂ are linearly independent. (below det means a 2 by 2 determinant)

y₁ y₂

WR = det ≠ 0

d y₁ /dx d y₂ /dx

NOTE: A second order d.e. requires two initial conditions in order to evaluate the

two constants of integration when integrating the d.e.

y ’’ + p(x) y ’ + q(x) y = 0

The initial conditions may appear as y(a) = c, y ’(a) = d, where c and d are constants

Click here to continue with discussion of the basics of second order, linear, ordinary d.e.’s.

Return to Differential Equations Notes

Copyright © 2017 Richard C. Coddington

All rights reserved.