11thhour Exam Mastery Notes Generator

### Differential Equations in Engineering Calculus: Master Notes for Exam Success --- ## **Introduction to Differential Equations** A **differential equation** (DE) is an equation that involves a function and its derivatives. These equations describe physical phenomena like motion, heat flow, electrical circuits, and more. ### **Key Terms to Know** - **Order**: The highest derivative in the equation (e.g., \( \frac{d^2y}{dx^2} \) makes it second-order). - **Degree**: The power of the highest derivative after clearing fractions and radicals (e.g., \( (\frac{dy}{dx})^2 \) has a degree of 2). - **Linear DE**: No product of the dependent variable and its derivatives (e.g., \( \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0 \)). - **Nonlinear DE**: Contains products or powers of the dependent variable or its derivatives. --- ## **Types of Differential Equations** 1. **Ordinary Differential Equations (ODEs)**: Involve one independent variable (e.g., time). Example: \( \frac{dy}{dx} + 3y = 0 \). 2. **Partial Differential Equations (PDEs)**: Involve multiple independent variables. Example: \( \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0 \). --- ## **First-Order Differential Equations** ### **Form**: \[ \frac{dy}{dx} = f(x, y) \] #### **Methods to Solve First-Order ODEs** 1. **Separable Variables**: - Form: \( \frac{dy}{dx} = g(x)h(y) \). - Steps: 1. Rearrange as \( \frac{1}{h(y)}dy = g(x)dx \). 2. Integrate both sides. 3. Solve for \( y \). **Example**: Solve \( \frac{dy}{dx} = xy \). - Rearrange: \( \frac{1}{y}dy = xdx \). - Integrate: \( \ln|y| = \frac{x^2}{2} + C \). - Solution: \( y = e^{\frac{x^2}{2} + C} = Ce^{\frac{x^2}{2}} \). 2. **Linear First-Order ODE**: - Form: \( \frac{dy}{dx} + P(x)y = Q(x) \). - Use **integrating factor (IF)**: \[ IF = e^{\int P(x)dx} \] - Solution: \[ y \cdot IF = \int Q(x) \cdot IF \, dx \] **Example**: Solve \( \frac{dy}{dx} + y = e^x \). - \( P(x) = 1 \), \( Q(x) = e^x \). - \( IF = e^{\int 1dx} = e^x \). - Solution: \( y \cdot e^x = \int e^x \cdot e^x dx = \int e^{2x}dx = \frac{e^{2x}}{2} + C \). - Final: \( y = \frac{e^x}{2} + Ce^{-x} \). --- ## **Second-Order Linear Differential Equations** ### **Form**: \[ a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x) \] - **Homogeneous**: \( f(x) = 0 \). - **Non-Homogeneous**: \( f(x) \neq 0 \). --- ### **Homogeneous Case** #### **Solve Using the Characteristic Equation**: \[ ar^2 + br + c = 0 \] 1. **Distinct Real Roots** (\( r_1, r_2 \)): \[ y = C_1e^{r_1x} + C_2e^{r_2x} \] 2. **Repeated Roots** (\( r_1 = r_2 = r \)): \[ y = (C_1 + C_2x)e^{rx} \] 3. **Complex Roots** (\( r = \alpha \pm \beta i \)): \[ y = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x)) \] #### **Example**: Solve \( \frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0 \). - Characteristic Equation: \( r^2 - 5r + 6 = 0 \). - Roots: \( r = 2, 3 \). - Solution: \( y = C_1e^{2x} + C_2e^{3x} \). --- ### **Non-Homogeneous Case** #### **Method of Undetermined Coefficients**: 1. Solve the complementary equation (\( f(x) = 0 \)). 2. Assume a form for the particular solution \( y_p \). 3. Substitute \( y_p \) into the DE to find coefficients. #### **Example**: Solve \( \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = e^x \). - Complementary Solution: \( r^2 + 3r + 2 = 0 \), roots \( r = -1, -2 \). \( y_c = C_1e^{-x} + C_2e^{-2x} \). - Particular Solution: Assume \( y_p = Ae^x \). Substitute: \( A(e^x) + 3A(e^x) + 2A(e^x) = e^x \). Solve: \( 6A = 1 \), so \( A = \frac{1}{6} \). - General Solution: \( y = y_c + y_p = C_1e^{-x} + C_2e^{-2x} + \frac{1}{6}e^x \). --- ## **Practice Questions** 1. Solve \( \frac{dy}{dx} = \frac{x}{y} \). 2. Solve \( \frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0 \). 3. Solve \( \frac{dy}{dx} + 2y = x^2 \). 4. Solve \( \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = e^x \). --- ## **Quick Revision Guide** - **Must-Know Formulas**: - \( IF = e^{\int P(x)dx} \). - Homogeneous solutions depend on the characteristic equation. - Non-homogeneous: \( y = y_c + y_p \). - **Key Tips**: - Always simplify the DE first. - For repeated roots, multiply \( x \) with \( e^{rx} \). - Use guessing methods (polynomials, exponentials, trigonometric functions) for \( y_p \). --- Let me know if you need solutions to the practice questions!