MA 262 Section 1.1: Introduction to Differential Equations

1. Basic Definitions

Ordinary Differential Equation (ODE)

An ODE involves one independent variable, some dependent variable/function, and its derivatives.

Examples of ODEs:

$y' = ky$ (where $y' = \frac{dy}{dx}$ and $k$ is a constant)
$x'' - 3x' + x = \cos(t)$ (where $x' = \frac{dx}{dt}$)
$\frac{dP}{dt} = k \cdot P(M - P)$ (where $k$ and $M$ are constants)

Partial Differential Equation (PDE)

A PDE involves multiple independent variables and partial derivatives.

Example: Heat Equation

Let $u(x, t)$ represent heat at position $x$ and time $t$:

$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial t} = 0$$

2. Verifying Solutions

A differential equation is like a "template" or "model." You plug a function into it to check if the function satisfies the equation (i.e., the equality is TRUE).

Example 1: Checking $y' = 7y$

Case A: Is $y = \sin(7x)$ a solution?

Derivative: $y' = 7\cos(7x)$

Plug into ODE: $7\cos(7x) \stackrel{?}{=} 7\sin(7x)$

Result: False. $y = \sin(7x)$ does not satisfy the ODE.

Case B: Is $y = e^{7x}$ a solution?

Derivative: $y' = 7e^{7x}$

Plug into ODE: $7e^{7x} \stackrel{?}{=} 7(e^{7x})$

Result: True. $y = e^{7x}$ does satisfy the ODE.

Example 2: Checking $y' = 12x^2(y^2 + 1)$

Case A: Is $y = e^x$ a solution?

Plug in: $e^x \stackrel{?}{=} 12x^2((e^x)^2 + 1)$

Result: No, $y = e^x$ is not a solution.

Case B: Is $y = \tan(4x^3 + C)$ a solution?

Derivative: $y' = \sec^2(4x^3 + C) \cdot (12x^2)$

Plug into ODE: $12x^2 \sec^2(4x^3 + C) \stackrel{?}{=} 12x^2(\tan^2(4x^3 + C) + 1)$

Using the identity $\sec^2(x) = \tan^2(x) + 1$:

Result: Yes, it is a solution.

3. General Solutions

Differential equations can have many solutions. A general solution describes an infinite family of solutions and usually involves an unknown constant $C$.

Examples:

For $y' = 2x$, the general solution is $y = x^2 + C$. (e.g., $y = x^2$, $y = x^2 - 3$, $y = x^2 + \pi$ are all individual solutions).
For $y' = 12x^2(y^2 + 1)$, the general solution is $y = \tan(4x^3 + C)$.
For $y' = 7y$, the general solution is $y = Ce^{7x}$.

Application: Newton's 2nd Law

In physics, if $x(t)$ describes the motion of an object, it must satisfy the differential equation:

$$m x'' = F$$

Section 1.1 Introduction to Differential Equations