Contents ix
42.3 Further worked problems on the t = 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
x n e x dx
426
44.3 Using reduction formulae for integrals of
the form
x n cos x dx and
x n sin x dx
427
44.4 Using reduction formulae for integrals of
the form
sin n x dx and
cos n x dx
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 form P
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 form a
d 2 y
dx 2 + b
dy
dx
+ cy = 0
478
50.3 Worked problems on differential equations
of the form a
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 form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
483
51.3 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is a constant or polynomial
484
51.4 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is an exponential function
486
51.5 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is a sine or cosine function
488
42.3 Further worked problems on the t = 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
x n e x dx
426
44.3 Using reduction formulae for integrals of
the form
x n cos x dx and
x n sin x dx
427
44.4 Using reduction formulae for integrals of
the form
sin n x dx and
cos n x dx
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 form P
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 form a
d 2 y
dx 2 + b
dy
dx
+ cy = 0
478
50.3 Worked problems on differential equations
of the form a
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 form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
483
51.3 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is a constant or polynomial
484
51.4 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is an exponential function
486
51.5 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is a sine or cosine function
488