Essential formulae 669
Simpson’s rule
ydx ≈
1
3
width of
interval
first + last
ordinate
+ 4
sum of even
ordinates
+ 2
sum of remaining
odd ordinates
Differential Equations
First order differential equations:
Separation of variables
If
d y
dx
= f (x) then y =
f (x) dx
If
d y
dx
= f (y) then
dx =
d y
f (y)
If
d y
dx
= f (x) · f (y) then
d y
f (y)
=
f (x) dx
Homogeneous equations:
If P
d y
dx
= Q, where P and Q are functions of both x and
y of the same degree throughout (i.e. a homogeneous
first order differential equation) then:
(i) Rearrange P
d y
dx
= Q into the form
d y
dx
=
Q
P
(ii) Make the substitution y = vx (where v is a function of x), from which, by the product rule,
d y
dx
= v(1) + x
dv
dx
(iii) Substitute for both y and
d y
dx
in the equation
d y
dx
=
Q
P
(iv) Simplify, by cancelling, and then separate the
variables and solve using the
d y
dx
= f (x) · f (y)
method
(v) Substitute v =
y
x
to solve in terms of the original
variables.
Linear first order:
If
d y
dx
+ P y = Q, where P and Q are functions of x
only (i.e. a linear first order differential equation), then
(i) determine the integrating factor, e
P dx
(ii) substitute the integrating factor (I.F.) into
the equation
y (I.F.) =
(I.F.) Q dx
(iii) determine the integral
(I.F.) Q dx
Numerical solutions of first order
differential equations:
Euler’s method:
y 1 = y 0 + h(y ) 0
Euler-Cauchy method: y P 1 = y 0 + h(y ) 0
and
y C 1 = y 0 +
1
2
h[(y ) 0 + f (x 1 , y p 1 )]
Runge-Kutta method:
To solve the differential equation
d y
dx
= f (x, y) given
the initial condition y = y 0 at x = x 0 for a range of
values of x = x 0 (h)x n :
1. Identify x 0 , y 0 and h, and values of x 1 , x 2 , x 3 , ...
2. Evaluate k 1 = f (x n , y n ) starting with n = 0
3. Evaluate k 2 = f
x n +
h
2
, y n +
h
2
k 1
4. Evaluate k 3 = f
x n +
h
2
, y n +
h
2
k 2
5. Evaluate k 4 = f(x n + h, y n + hk 3 )
6. Use the values determined from steps 2 to 5 to
evaluate:
y n+1 = y n +
h
6
{k 1 + 2k 2 + 2k 3 + k 4 }
7. Repeat steps 2 to 6 for n = 1, 2, 3, ...
Second order differential equations:
If a
d 2 y
dx 2 + b
dy
dx
+ cy = 0 (where a, b and c are constants) then:
(i) rewrite the differential equation as
(aD 2 + bD +c)y = 0
(ii) substitute m for D and solve the auxiliary equation
am 2 + bm + c = 0
Simpson’s rule
ydx ≈
1
3
width of
interval
first + last
ordinate
+ 4
sum of even
ordinates
+ 2
sum of remaining
odd ordinates
Differential Equations
First order differential equations:
Separation of variables
If
d y
dx
= f (x) then y =
f (x) dx
If
d y
dx
= f (y) then
dx =
d y
f (y)
If
d y
dx
= f (x) · f (y) then
d y
f (y)
=
f (x) dx
Homogeneous equations:
If P
d y
dx
= Q, where P and Q are functions of both x and
y of the same degree throughout (i.e. a homogeneous
first order differential equation) then:
(i) Rearrange P
d y
dx
= Q into the form
d y
dx
=
Q
P
(ii) Make the substitution y = vx (where v is a function of x), from which, by the product rule,
d y
dx
= v(1) + x
dv
dx
(iii) Substitute for both y and
d y
dx
in the equation
d y
dx
=
Q
P
(iv) Simplify, by cancelling, and then separate the
variables and solve using the
d y
dx
= f (x) · f (y)
method
(v) Substitute v =
y
x
to solve in terms of the original
variables.
Linear first order:
If
d y
dx
+ P y = Q, where P and Q are functions of x
only (i.e. a linear first order differential equation), then
(i) determine the integrating factor, e
P dx
(ii) substitute the integrating factor (I.F.) into
the equation
y (I.F.) =
(I.F.) Q dx
(iii) determine the integral
(I.F.) Q dx
Numerical solutions of first order
differential equations:
Euler’s method:
y 1 = y 0 + h(y ) 0
Euler-Cauchy method: y P 1 = y 0 + h(y ) 0
and
y C 1 = y 0 +
1
2
h[(y ) 0 + f (x 1 , y p 1 )]
Runge-Kutta method:
To solve the differential equation
d y
dx
= f (x, y) given
the initial condition y = y 0 at x = x 0 for a range of
values of x = x 0 (h)x n :
1. Identify x 0 , y 0 and h, and values of x 1 , x 2 , x 3 , ...
2. Evaluate k 1 = f (x n , y n ) starting with n = 0
3. Evaluate k 2 = f
x n +
h
2
, y n +
h
2
k 1
4. Evaluate k 3 = f
x n +
h
2
, y n +
h
2
k 2
5. Evaluate k 4 = f(x n + h, y n + hk 3 )
6. Use the values determined from steps 2 to 5 to
evaluate:
y n+1 = y n +
h
6
{k 1 + 2k 2 + 2k 3 + k 4 }
7. Repeat steps 2 to 6 for n = 1, 2, 3, ...
Second order differential equations:
If a
d 2 y
dx 2 + b
dy
dx
+ cy = 0 (where a, b and c are constants) then:
(i) rewrite the differential equation as
(aD 2 + bD +c)y = 0
(ii) substitute m for D and solve the auxiliary equation
am 2 + bm + c = 0