The Chemistry Maths Book, Second Edition

330 Chapter 11First-order differential equations Therefore, by formula (11.53), 0 Exercises 33–40 11.6 An example of linear equa ...

11.6 An example of linear equations in chemical kinetics 331 or (11.59) This is a linear equation withp(t) 1 = 1 k 2 ,r(t) 1 = 1 ...

332 Chapter 11First-order differential equations amount of intermediate Bremains small throughout the reaction because Bis rapid ...

11.7 Electric circuits 333 whereLis the inductance. A simple capacitor is made up of two parallel metallic plates separated by a ...

334 Chapter 11First-order differential equations 11.8 Exercises Section 11.2 State the order of the differential equation and ve ...

11.8 Exercises 335 Use the method in Exercise 26 to find the general solution: Section 11.4 29.Find the intervalτ 12 n i ...

336 Chapter 11First-order differential equations where(a 1 − 1 x),y, zand uare the amounts of A, B, C, and D, respectively, at t ...

12 Second-order differential equations. Constant coefficients 12.1 Concepts The second-order differential equations that are imp ...

338 Chapter 12Second-order differential equations. Constant coefficients EXAMPLE 12.1Show thaty 1 1 = 1 e 2 x andy 2 1 = 1 e 3 x ...

12.2 Homogeneous linear equations 339 so thaty 3 1 = 1 cy 1 is also a solution. The functionsy 1 andy 3 are not regarded as dist ...

340 Chapter 12Second-order differential equations. Constant coefficients EXAMPLE 12.2The general solution of the equation in Exa ...

12.3 The general solution 341 and the possible solutions of type (12.6) are (12.9) EXAMPLE 12.3The characteristic equation of th ...

342 Chapter 12Second-order differential equations. Constant coefficients (ii) A real double root When a 2 1 − 14 b 1 = 10 the tw ...

12.3 The general solution 343 (iii) Complex roots When the discriminant a 2 1 − 14 bof the characteristic quadratic equation (12 ...

344 Chapter 12Second-order differential equations. Constant coefficients where(a 2 2) 2 1 − 1 b 1 < 10. Let(a 2 2) 2 1 − 1 b ...

12.4 Particular solutions 345 EXAMPLE 12.6Solve the initial value problem y′′ 1 + 1 y′ 1 − 16 y 1 = 1 0, y(0) 1 = 1 0, y′(0) 1 = ...

346 Chapter 12Second-order differential equations. Constant coefficients Application of the boundary conditions gives y(0) 1 = 1 ...

12.4 Particular solutions 347 or by the equivalent trigonometric form (12.17), y(x) 1 = 1 d 1 cos 1 ωx 1 + 1 d 2 1 sin 1 ωx (12. ...

348 Chapter 12Second-order differential equations. Constant coefficients Then y(θ 1 + 1 π) 1 = 1 c 1 e i ω θ e i ωπ 1 + 1 c 2 e ...

12.5 The harmonic oscillator 349 EXAMPLE 12.10The vibrations of diatomic molecules The vibrations of a diatomic molecule are oft ...