Unit 3: Calculus (Part 3)
Differential Equations & Modeling [Syllabus 3.10 – 3.11]
1. Basic Concepts
| Differential Equation (D.E.) | An equation involving independent variables, dependent variables, and the derivatives of the dependent variable with respect to the independent variable. |
| Order | The order of the highest order derivative appearing in the equation.
Ex: $$\frac{d^2y}{dx^2} + y = 0$$ has Order = 2. |
| Degree | The highest power (exponent) of the highest order derivative when the equation is a polynomial in derivatives.
Note: Degree is not defined if the equation involves terms like $$\sin(\frac{dy}{dx})$$, $$e^{y’}$$, or $$\log(y’)$$. |
2. Formation of Differential Equations
Used to find the differential equation representing a family of curves.
| Procedure | 1. Count the number of arbitrary constants (parameters) in the given equation (say $$n$$). 2. Differentiate the equation $$n$$ times. 3. Eliminate the arbitrary constants using the original equation and the derivatives. |
| General vs. Particular Solution | General Solution: Contains arbitrary constants equal to the order of the D.E. Particular Solution: Obtained by giving specific values to the arbitrary constants (contains no arbitrary constants). |
3. Methods of Solving D.E.
Syllabus Scope: Simple differential equations and Variable Separable method only.
| Method 1: Direct Integration | If $$\frac{dy}{dx} = f(x)$$: Solution: $$y = \int f(x) dx + C$$ |
| Method 2: Variable Separable | If the equation can be written as $$f(x) dx = g(y) dy$$: Step 1: Separate x terms with dx and y terms with dy. Step 2: Integrate both sides: $$\int f(x) dx = \int g(y) dy + C$$. |
| Mathematical Modeling | Formulating a physical problem into a differential equation (e.g., Growth/Decay problems). Example: Rate of growth proportional to amount present: $$\frac{dy}{dt} = ky$$. |