A calculus of the absurd

14.4 Second-Order Differential Equations

14.4.1 Introduction

The easiest second-order differential equations to solve are those which we can integrate directly, for example

\begin{equation} \frac {d^2y}{dx^2} = \cos (x) \end{equation}

When we integrate this once, we get that

\begin{align} & \int \frac {d^2y}{dx^2} dx = \int \cos (x) dx \\ & \frac {dy}{dx} = \sin (x) + c \end{align}

and then integrating again, we get that

\begin{align} & \int \frac {dy}{dx} dx = \int \sin (x) + c dx \\ & y(x) = -\cos (x) + cx + d \end{align}

Which is the general solution to this particular second-order differential equation. With first-order differential equations we only have one constant, and we can determine the value of the constant given a single point on the curve. 9797 In more formal notation, for the differential equation
\begin{equation*} \frac {dy}{dx} = f(x) \end{equation*}
with solution \(y(x) = F(x) + c\) we can determine the value of \(c\) given a value of \(x\), \(x_0\) and the corresponding value of \(y\), \(y_0\) at that point.

For a second-order differential equation, however, we have two constants, so we definitely can’t solve the equation given only point. We need either two points on the curve, or one point on the original curve and one point on its first derivative.