Ordinary Differential Equations Gabriel Nagy

First Order Equations We start our study of differential equations in the same way the pioneers in this field did. We show particular techniques to solve particular types of first order differential equations. The techniques were developed in the eighteenth and nineteenth centuries and the equations include linear equations, separable equations, Euler homogeneous equations, and exact equations. This way of studying differential equations reached a dead end pretty soon. Most of the differential equations cannot be solved by any of the techniques presented in the first sections of this chapter. People then tried something different. Instead of solving the equations they tried to show whether an equation has solutions or not, and what properties such solution may have. This is less information than obtaining the solution, but it is still valuable information. The results of these efforts are shown in the last sections of this chapter. We present theorems describing the existence and uniqueness of solutions to a wide class of first order differential equations. t y π 2 0 − π 2 y 0 = 2 cos(t) cos(y) 3 4 1. FIRST ORDER EQUATIONS 1.1. Linear Constant Coefficient Equations 1.1.1. Overview of Differential Equations. A differential equation is an equation, where the unknown is a function and both the function and its derivatives may appear in the equation. Differential equations are essential for a mathematical description of nature— they lie at the core of many physical theories. For example, let us just mention Newton’s and Lagrange’s equations for classical mechanics, Maxwell’s equations for classical electromagnetism, Schr¨odinger’s equation for quantum mechanics, and Einstein’s equation for the general theory of gravitation. We now show what differential equations look like. Example 1.1.1. (a) Newton’s law: Mass times acceleration equals force, ma = f, where m is the particle mass, a = d 2x/dt2 is the particle acceleration, and f is the force acting on the particle. Hence Newton’s law is the differential equation m d 2x dt2 (t) = f  t, x(t), dx dt (t)  , where the unknown is x(t)—the position of the particle in space at the time t. As we see above, the force may depend on time, on the particle position in space, and on the particle velocity. Remark: This is a second order Ordinary Differential Equation (ODE). (b) Radioactive Decay: The amount u of a radioactive material changes in time as follows, du dt (t) = −k u(t), k 0, where k is a positive constant representing radioactive properties of the material. Remark: This is a first order ODE. (c) The Heat Equation: The temperature T in a solid material changes in time and in three space dimensions—labeled by x = (x, y, z)—according to the equation ∂T ∂t (t, x) = k ∂ 2T ∂x2 (t, x) + ∂ 2T ∂y2 (t, x) + ∂ 2T ∂z2 (t, x)  , k 0, where k is a positive constant representing thermal properties of the material. Remark: This is a first order in time and second order in space PDE. (d) The Wave Equation: A wave perturbation u propagating in time t and in three space dimensions—labeled by x = (x, y, z)—through the media with wave speed v 0 is ∂ 2u ∂t2 (t, x) = v 2 ∂ 2u ∂x2 (t, x) + ∂ 2u ∂y2 (t, x) + ∂ 2u ∂z2 (t, x)  . Remark: This is a second order in time and space Partial Differential Equation (PDE). C The equations in examples (a) and (b) are called ordinary differential equations (ODE)— the unknown function depends on a single independent variable, t. The equations in examples (d) and (c) are called partial differential equations (PDE)—the unknown function depends on two or more independent variables, t, x, y,