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3.1 Solutions of Non-Linear Equations
Solve the following equations :
1. Solve 0542 xx
2. Solve 083 x
3. Solve 03 xex
So, in this subtopic we are to going discuss numerical methods in
finding the approximate numerical solutions of such equations.
Introduction
Many equations cannot be solved exactly, but various methods of
finding approximate numerical solutions exist.
The most commonly used methods have two main parts:
(a) finding an initial approximate value(b) improving this value by an iterative process
Initial Values:
The initial value of the roots of f(x) = 0 can be located
approximately by either a graphical or an algebraic method.
GRAPHICAL METHOD:
Either
(a) Plot ( or sketch ) the graph ofy =f(x).
The real roots are the points where the curve cuts the x axis.
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Or
(b) Rewritef(x) = 0 in the form F(x) = G(x).
Plot ( or sketch ) y =F(x) andy = G(x). The real roots are at
the points where these graphs intersect.
Example 1
Find the approximate value of the equation 04ln xx by
using the graphical method.
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Example 2
On the same axes, sketch 1 xe y and .1 xy Show that the
equation01)1ln(
xx has only one root .The sketch shows that there is only one point of intersection and so
the equation 01)1ln( xx has only one root .
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Example 3
By sketching the graph of 3xy and 13 xy on the same
coordinate axes, show that the equation 0133
xx has threereal roots.
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Algebraic Method
Find two values a and b such that f(a) and f(b) have
different signs.
At least one root must lie between a and b if f(x) is
continuous.
If more than one root is suspected between a and b, sketch
y =f(x).
Example 4
Show that the equation 043 xx has a root between 3.1x
and 4.1x . Approximate the root to two decimal places.
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3.2 Iterative Method
All iterative methods follow the same basic pattern. A sequence of
approximations ,......,,, 4321 xxxx is found, each one closer to the
root of 0xf . Each approximation is found from the one
before it using a specified method. The process continue until the
required accuracy is reached.
Iteration Method
Rewrite the equation 0xf in the form xgx
If the initial approximation is ,1x then calculate
12 xgx
23 xgx
34 xgx
..
This method fails if 1' 1 xg near the root.
So we are looking for 1' 1 xg .
Example 5
Using the iteration method, find the solution of xexxf near
5.0x correct to three decimal places.
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Example 6
Show that the equation 0412
xx has a root between 0.2 and
0.3. Taking 0.2 as the first approximation, use the iteration method
to find the root of the equation. Give your answer correct to three
significant figures.
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Example 7
Show that the equation 05.0sin xx has a root lying in the
interval 1,2 . Using the iteration formula 5.0sin1 nn xx ,
find the root correct to three decimal places.
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Example 8
Given the equation 02ln xx
a) Show that there exist a real root between 1.5 and 1.6
b) Find the appropriate iterative function for the equation.
c) Hence, find the real root of the equation correct to three decimal
places by using iterative method.
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EXERCISE
1. Show that the equation 03 xex has a root between 0and 1. By taking the first approximation x1 as 0.8, use
iteration method to find the root correct to 3 decimal places.
(Ans: 0.792)
2. Find an approximate root of the equation .043 xx Find this root correct to 3 decimal places using iteration
method. (Ans: 1.379)
3. Sketch the graph of )2(ln xy where x
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3.3 Newton-Raphson method
Newton-Raphson Method
If 1x is the first approximation to the root of the equation
0xf , then second, third,. approximations are written as
,........,, 432 xxx and are given by the formula
11
12' xf
xfxx
Repeat this process as required
nn
nnxf
xfxx
'1
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Example 9
Using the Newton-Raphson Method, find the solution of xexxf
near 5.0x correct to three decimal places.
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Example 10
Show that the equation 063 xx has a root between 1 and 2.
Using the Newton-Raphson method with the starting point 1.6,
determine an approximation to this root, giving your answer to
three significant figures.
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Example 11
Use Newton-Raphson method to find an approximate value of 315
. Give your answer correct to three decimal places.
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Example 12
Sketch the graph of xey and y = 2x where x < 2, on the
same axes. Get the first approximation, 0x for the equation
ex
= 2 x where 0 < 0x < 1. Hence, by using Newton-Raphson
method , solve the equation ofx
ex
2
1for x < 2 to three
decimal places.
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Example 13
Show that the equation 0sin2 xx has a root between 1x
(radian) and 2x (radian). Find the root of equation by using
a)Iteration methodb)Newton Raphson method
giving your answer correct to two decimal places.