Introduction to Differential Equations

11.1 What is a Differential Equation?

Definition of a Differential Equation

A differential equation is an equation that relates an unknown function to one or more of its derivatives. These equations are fundamental in describing various physical, biological, and economic phenomena where change is involved.

• Unknown function: The function we are trying to find.

• Derivatives: Expressions representing rates of change of the unknown function.

Examples of Differential Equations

• $N'(t) = -kN(t)$

• $y' = y$

• $\frac{dy}{dx} = -6x y$

• $y''(t) = 6.25y$

• $y'' + 2y' + y = 4$

Classification:

• First-order differential equations: Involve only the first derivative (e.g., $y' = y$).

• Second-order differential equations: Involve the second derivative (e.g., $y''(t) = 6.25y$).

Solving Differential Equations: Example Problems

To determine if a function is a solution to a differential equation, substitute the function and its derivatives into the equation and check if the equality holds.

• Example 1: Show $y(t) = Ce^{2.5t}$ is a solution to $y'(t) = 2.5y(t)$.

  • Compute $y'(t)$: $y'(t) = C \cdot 2.5 e^{2.5t}$

  • Since $y(t) = Ce^{2.5t}$, $2.5y(t) = 2.5Ce^{2.5t}$

  • Therefore, $y'(t) = 2.5y(t)$

• Example 2: Show $y(x) = e^{2x}$ is not a solution to $\frac{d^2y}{dx^2} - 4y = 0$.

  • Compute derivatives: $y'(x) = 2e^{2x}$, $y''(x) = 4e^{2x}$

  • Substitute: $y'' - 4y = 4e^{2x} - 4e^{2x} = 0$

  • But the note says $y'' - 4y = 8e^{2x}$, which is not zero for all $x$ (Additional info: There may be a typo in the original notes; the correct calculation is $0$.)

Initial Value Problems (IVPs)

A initial value problem (IVP) consists of a differential equation together with an initial condition specifying the value of the unknown function at a given point.

• Differential equation: $y'(t) = 2.5y(t)$

• Initial value: $y(0) = 3.2$

Together, these form an IVP.

General and Particular Solutions

• General solution: The family of all possible solutions, typically containing an arbitrary constant (e.g., $y(t) = Ce^{2.5t}$).

• Particular solution: A specific solution found by applying the initial condition to determine the constant.

Example: Given $y(0) = 3.2$, solve for $C$:

• $y(0) = Ce^{2.5 \cdot 0} = C \cdot 1 = C$

• So $C = 3.2$

• Therefore, $y(t) = 3.2e^{2.5t}$

Additional info: Differential equations are widely used in modeling real-world systems, such as population growth, radioactive decay, and mechanical vibrations.