Mathematics · Ch 9 — Differential Equations

Summary

Order and Degree: Order is the highest derivative present; degree is the power of the highest derivative after removing radicals and fractions.
Formation of DE: Eliminate arbitrary constants from the family of curves to obtain the differential equation.
Solution types:
- General solution: Contains arbitrary constants equal to the order.
- Particular solution: Obtained by substituting initial/boundary conditions.
Variable Separable: Write as $g(y) \, dy = f(x) \, dx$ , then integrate both sides.
Homogeneous DE: Of the form $\frac{dy}{dx} = F\left(\frac{y}{x}\right)$ . Substitute $y = vx$ , then separate variables.
Linear DE: $\frac{dy}{dx} + P(x) y = Q(x)$ . Integrating factor $I.F. = e^{\int P \, dx}$ . Solution: $y \cdot I.F. = \int Q \cdot I.F. \, dx + C$ . …