Differential equations are mathematical models that describe how quantities change over time or space. They're crucial in physics, engineering, and other sciences for predicting system behavior and solving real-world problems.

Understanding differential equations involves grasping their order, types, and solutions. We'll explore how to classify equations, find general and particular solutions, and solve initial-value problems. These concepts form the foundation for tackling more complex differential equations.

Differential Equations Fundamentals

Order of differential equations

• Determined by the highest derivative present in the equation
  • $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} - y = 0$ is a third-order differential equation due to $\frac{d^3y}{dx^3}$
• Dictates the number of initial conditions required to solve the equation
  • First-order equations need one initial condition ($y(x_0) = y_0$)
  • Second-order equations need two initial conditions ($y(x_0) = y_0$ and $y'(x_0) = y'_0$)
  • Third-order equations need three initial conditions ($y(x_0) = y_0$, $y'(x_0) = y'_0$, and $y''(x_0) = y''_0$)

Types of Differential Equations

• Ordinary differential equations (ODEs) involve functions of a single independent variable and their derivatives
• Partial differential equations (PDEs) involve functions of multiple independent variables and their partial derivatives
• Separable equations are ODEs where variables can be separated and integrated independently
• Linear equations have a specific form where the dependent variable and its derivatives appear linearly
• Homogeneous equations are a special case of linear equations where all terms contain the dependent variable or its derivatives

Solutions to differential equations

• Function that satisfies the equation for all values of the independent variable within the domain
  • Substituting the solution function into the differential equation results in an identity
• Can be expressed as explicit functions ($y = f(x)$), implicit functions ($F(x, y) = 0$), or parametric equations ($x = x(t)$, $y = y(t)$)
• Domain of the solution must be considered to ensure validity for all relevant values of the independent variable
  • Solution $y = \sqrt{1 - x^2}$ to $\frac{dy}{dx} = -\frac{x}{\sqrt{1 - x^2}}$ is only valid for $-1 \leq x \leq 1$

General vs particular solutions

• General solution contains all possible solutions to a differential equation
  • Includes arbitrary constants ($C_1$, $C_2$, etc.) that can take on any value
  • $y = C_1\cos(x) + C_2\sin(x)$ is a general solution to $\frac{d^2y}{dx^2} + y = 0$
• Particular solution is a specific solution obtained from the general solution by assigning values to the arbitrary constants
  • Satisfies the differential equation and given initial or boundary conditions
  • $y = 2\cos(x) - 3\sin(x)$ is a particular solution to $\frac{d^2y}{dx^2} + y = 0$ with $y(0) = 2$ and $y'(0) = -3$
• For linear equations, the characteristic equation helps determine the general solution

Initial-value problems and significance

• Differential equation paired with one or more initial conditions
  • Initial conditions specify the value of the function and/or its derivatives at a specific point (usually $x = 0$ or $t = 0$)
  • $y' = y$ with $y(0) = 1$ is an initial-value problem
• Allow finding a unique particular solution to a differential equation
  • Number of initial conditions needed depends on the order of the differential equation
• Solving an initial-value problem involves finding a particular solution that satisfies both the differential equation and the given initial conditions
  • Solution to $y' = y$ with $y(0) = 1$ is $y = e^x$
• For certain types of equations, an integrating factor can be used to solve initial-value problems

Verification of differential equation solutions

• To verify if a function satisfies a differential equation:
  1. Substitute the function into the differential equation
  2. Perform the necessary derivatives and simplify the equation
  3. Check if the resulting equation is an identity (true for all values of the independent variable)
• To verify if a function satisfies an initial-value problem:
  1. Check if the function satisfies the differential equation (using the steps above)
  2. Evaluate the function and/or its derivatives at the specified initial condition(s)
  3. Confirm that the values obtained in step 2 match the given initial conditions
• Example: Verify that $y = e^x$ satisfies $y' = y$ with $y(0) = 1$
  1. $y' = e^x = y$, so $y = e^x$ satisfies the differential equation
  2. $y(0) = e^0 = 1$, matching the initial condition