Practical examples of differential equation classification (Order, Linearity, Type).

📚 Quick Study Guide

• 🔢 Order: The order of a differential equation is the highest derivative present in the equation. For example, $y'' + y' + y = 0$ is a second-order differential equation.
• 📈 Linearity: A differential equation is linear if it can be written in the form $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... + a_1(x)y' + a_0(x)y = f(x)$, where the coefficients $a_i(x)$ are functions of $x$ only. Non-linear terms include $y^2$, $\sin(y)$, or $y y'$.
• 🔍 Type: Differential equations can be classified as Ordinary Differential Equations (ODEs) or Partial Differential Equations (PDEs). ODEs involve functions of a single independent variable, while PDEs involve functions of multiple independent variables.

🧪 Practice Quiz

1. What is the order of the differential equation $y''' + 2y'' - y' + y = x^2$?
   1. 1
   2. 2
   3. 3
   4. 4
2. Is the differential equation $y'' + \sin(y) = 0$ linear or non-linear?
   1. Linear
   2. Non-linear
   3. Cannot be determined
   4. Both
3. Which type of differential equation is $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$?
   1. Ordinary Differential Equation (ODE)
   2. Partial Differential Equation (PDE)
   3. Algebraic Equation
   4. Integral Equation
4. What is the order of the differential equation $\frac{dy}{dx} + y = e^x$?
   1. 0
   2. 1
   3. 2
   4. 3
5. Is the differential equation $x^2y'' + xy' + y = \cos(x)$ linear or non-linear?
   1. Linear
   2. Non-linear
   3. Cannot be determined
   4. Both
6. Which type of differential equation is $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$?
   1. Partial Differential Equation (PDE)
   2. Ordinary Differential Equation (ODE)
   3. Algebraic Equation
   4. Transcendental Equation
7. Is the differential equation $y' + y^2 = x$ linear or non-linear?
   1. Linear
   2. Non-linear
   3. Cannot be determined
   4. Both