Higher Engineering Mathematics, Sixth Edition

Contents ix

42.3 Further worked problems on thet=tan

θ

2

substitution 416

Revision Test 12 419

43 Integration by parts 420
43.1 Introduction 420
43.2 Worked problems on integration by parts 420
43.3 Further worked problems on integration
by parts 422

44 Reduction formulae 426
44.1 Introduction 426
44.2 Using reduction formulae for integrals of
the form

∫

xnexdx 426

44.3 Using reduction formulae for integrals of

the form

∫

xncosxdxand

∫

xnsinxdx 427

44.4 Using reduction formulae for integrals of

the form

∫

sinnxdxand

∫

cosnxdx 429

44.5 Further reduction formulae 432

45 Numerical integration 435
45.1 Introduction 435
45.2 The trapezoidal rule 435
45.3 The mid-ordinate rule 437
45.4 Simpson’s rule 439

Revision Test 13 443

46 Solution of first order differential equations by
separation of variables 444
46.1 Family of curves 444
46.2 Differential equations 445
46.3 The solution of equations of the form
dy
dx

=f(x) 445

46.4 The solution of equations of the form

dy

dx

=f(y) 447

46.5 The solution of equations of the form

dy

dx

=f(x)·f(y) 449

47 Homogeneous first order differential equations 452
47.1 Introduction 452
47.2 Procedure to solve differential equations
of the formP

dy

dx

=Q 452

47.3 Worked problems on homogeneous first

order differential equations 452

47.4 Further worked problems on homogeneous

first order differential equations 454

48 Linear first order differential equations 456

48.1 Introduction 456

48.2 Procedure to solve differential equations

of the form

dy

dx

+Py=Q 457

48.3 Worked problems on linear first order

differential equations 457

48.4 Further worked problems on linear first

order differential equations 458

49 Numerical methods for first order differential

equations 461

49.1 Introduction 461

49.2 Euler’s method 461

49.3 Worked problems on Euler’s method 462

49.4 An improved Euler method 466

49.5 The Runge-Kutta method 471

Revision Test 14 476

50 Second order differential equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy= 0 477

50.1 Introduction 477

50.2 Procedure to solve differential equations

of the forma

d^2 y

dx^2

+b

dy

dx

+cy= 0 478

50.3 Worked problems on differential equations

of the forma

d^2 y

dx^2

+b

dy

dx

+cy= 0 478

50.4 Further worked problems on practical

differential equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy= 0 480

51 Second order differential equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy=f(x) 483

51.1 Complementary function and particular

integral 483

51.2 Procedure to solve differential equations

of the forma

d^2 y

dx^2

+b

dy

dx

+cy=f(x) 483

51.3 Worked problems on differential equations

of the forma

d^2 y

dx^2

+b

dy

dx

+cy=f(x)

wheref(x)is a constant or polynomial 484

51.4 Worked problems on differential equations

of the forma

d^2 y

dx^2

+b

dy

dx

+cy=f(x)

wheref(x)is an exponential function 486

51.5 Worked problems on differential equations

of the forma

d^2 y

dx^2

+b

dy

dx

+cy=f(x)

wheref(x)is a sine or cosine function 488