Higher Engineering Mathematics, Sixth Edition

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

dy

dx

=f(x) theny=

∫

f(x)dx

If

dy

dx

=f(y) then

∫

dx=

∫

dy

f(y)

If

dy

dx

=f(x)·f(y) then

∫

dy

f(y)

=

∫

f(x)dx

Homogeneous equations:

IfP

dy

dx

=Q,wherePandQare functionsof bothxand
yof the same degree throughout (i.e. a homogeneous
first order differential equation) then:

(i) RearrangeP

dy

dx

=Qinto the form

dy

dx

=

Q

P

(ii) Make the substitutiony=vx(wherevis a func-

tion ofx), from which, by the product rule,

dy

dx

=v( 1 )+x

dv

dx

(iii) Substitute for bothy and

dy

dx

in the equation

dy

dx

=

Q

P

(iv) Simplify, by cancelling, and then separate the

variables and solve using the

dy

dx

=f(x)·f(y)

method

(v) Substitutev=

y

x

to solve in terms of the original

variables.

Linear first order:

If

dy

dx

+Py=Q,wherePandQare functions ofx

only (i.e. a linear first order differential equation), then

(i) determine the integrating factor, e

∫

Pdx

(ii) substitute the integrating factor (I.F.) into

the equation

y(I.F.)=

∫

(I.F.)Qdx

(iii) determine the integral

∫

(I.F.)Qdx

Numerical solutions of first order

differential equations:

Euler’s method: y 1 =y 0 +h(y′) 0

Euler-Cauchy method: yP 1 =y 0 +h(y′) 0

and yC 1 =y 0 +

1

2

h[(y′) 0 +f(x 1 ,yp 1 )]

Runge-Kutta method:

To solve the differential equation

dy

dx

=f(x,y)given

the initial conditiony=y 0 atx=x 0 for a range of

values ofx=x 0 (h)xn:

1. Identifyx 0 ,y 0 andh, and values ofx 1 ,x 2 ,x 3 ,...


2. Evaluatek 1 =f(xn,yn)starting withn= 0


3. Evaluatek 2 =f

(

xn+

h

2

,yn+

h

2

k 1

)

4. Evaluatek 3 =f

(

xn+

h

2

,yn+

h

2

k 2

)

5. Evaluatek 4 =f(xn+h,yn+hk 3 )


6. Use the values determined from steps 2 to 5 to
   evaluate:

yn+ 1 =yn+

h

6

{k 1 + 2 k 2 + 2 k 3 +k 4 }

7. Repeat steps 2 to 6 forn= 1 , 2 , 3 ,...

Second order differential equations:

Ifa

d^2 y

dx^2

+b

dy

dx

+cy= 0 (wherea,bandcare con-

stants) then:

(i) rewrite the differential equation as

(aD^2 +bD+c)y= 0

(ii) substitutemforDandsolvetheauxiliaryequation

am^2 +bm+c= 0