Differential equations appeared earlier in this book, principally in three sections. In contrast, 2 cos x + 10 is called a particular solution of the equation. We call 2 cos x + C the general solution of the given equation, since it can be shown that every solution can be written in this form. Thus, f(x) = 2 cos x + 10 is a solution to y + 2 sin x = 0 since f(x) + 2 sin x = -2 sin x + 2 sin x = 0 for all x.
If, when f(x) is substituted for y in the differential equation, the resulting equation is an identity for all x in some interval, then f(x) is called a solution of the differential equation. Examples of differential equations of orders 1, 2, and 3 are y + 2 sin x = 0 dy dx dx2 2ġ5.1 Linear Homogeneous Equations 15.2 Nonhomogeneous Equations 15.3 Applications of Second-Order Equations
In particular, an equation of the form F (x, y, y(1), y(2),, y(n)) = 0 in which y(k) denotes the kth derivative of y with respect to x, is called an ordinary differential equation of order n.
Differential EquationsLinear Homogeneous EquationsWe call an equation involving one or more derivatives of an unknown function a differential equation.
