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Chapter Analysis
Advanced40 pages • EnglishQuick Summary
This chapter on Differential Equations introduces the concept of differential equations, their order, degree, and methods to solve them. It covers solutions to first order first degree differential equations using methods like separation of variables and homogeneous equations. The chapter also discusses using integrating factor for linear differential equations and finding particular solutions given initial conditions.
Key Topics
- •Order of differential equations
- •Degree of differential equations
- •Methods of solving first order differential equations
- •Homogeneous differential equations
- •Integrating factor for linear equations
Learning Objectives
- ✓Understand the order and degree of differential equations.
- ✓Solve first order linear differential equations using integrating factors.
- ✓Apply variable separable technique to solve appropriate differential equations.
- ✓Identify and solve homogeneous differential equations.
Questions in Chapter
For each of the differential equations given in Exercises 1 to 12, find the general solution: 1. 2 sin dy/y x dx + = 2. 3 x dy/y e dx −+ = 3. 2 dy y / x dx x + = ...
Page 329
For each of the differential equations in Exercises 13 to 15, find a particular solution satisfying the given condition: 13. 2 tan sin ; dy/y x x π dx + = 0 when x = π/3 ...
Page 335
Additional Practice Questions
Solve the differential equation y'' + 3y' + 2y = 0.
mediumAnswer: The characteristic equation is r^2 + 3r + 2 = 0. Solving this gives r = -1, -2; the general solution is y(x) = C1 e^(-x) + C2 e^(-2x).
Find the solution of dy/dx = x^2 + y^2, given y(0) = 1.
hardAnswer: This equation cannot be solved by simple methods as it is a Riccati equation. Numerical methods or transformations could be required for an explicit solution.
Determine the order and degree of the differential equation (d^3y/dx^3)^2 + (dy/dx)^4 = 0.
easyAnswer: The order of the differential equation is 3, and the degree is 2, as the highest derivative raised to a power is (d^3y/dx^3)^2.