Transcript [email protected] Contents Solution of Algebraic Equations • • • • Polynomial to symbolic expression Symbolic expression to polynomial conversion Solution of algebraic equation Solution of system of linear equations Taylor.

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[email protected]
Contents
Solution of Algebraic Equations
•
•
•
•
Polynomial to symbolic expression
Symbolic expression to polynomial conversion
Solution of algebraic equation
Solution of system of linear equations
Taylor Series and Function Calculator
• Taylor Series Calculator
• Function Calculator
Solution of Ordinary Differential Equations
•
•
•
•
Ordinary differential Equations
Solution of ordinary Differential Equations
Solution of Non-linear Differential Equations
Solution of several Differential Equations
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Polynomial to Symbolic Conversion
• If you have stored a polynomial as a coefficient matrix [1 -3 2 5].
• To Convert this into symbolic polynomial we can use poly2sym
command.
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Symbolic to Polynomial Conversion
• To Convert symbolicexpression to numeric polynomial we can
use MATLAB function sym2poly.
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Functional Inverse
• g = finverse(f) returns the functional inverse of f.
• g = finverse(f,v) uses the symbolic variable v, where v is a sym,
as the independent variable.
returns
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Exercise#1
1. Convert following polynomial to symbolic expression.
V  1
3
0
1
2. Convert symbolic expression to polynomial
f  2t  3t  t 10
4
2
3. Find inverse of the function
f  sin(x)
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Solution of Algebraic Equations
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Solving Algebraic Equations
If S is a symbolic expression,
attempts to find values of the symbolic variable in S for which S is zero.
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Solving Algebraic Equations
• If you want to solve for a specific variable, you must specify that variable
as an additional argument.
• For example, if you want to solve S for b, use the command
which returns
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Solving Algebraic Equations
• Note that these equations are of the form f(x)=0.
• If you need to solve the equations of the form f(x)=q(x), you must use
quoted string.
• For example the command
which returns
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Exercise#2
1. Solve following equation.
f  ax  3x  bx  4
3
2
2. Solve for y
f  x  2 xy  y
2
2
3. Solve following equation for t.
t 2  3t  4  6
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Several Algebraic Equations
Suppose you have the system
x2 y2  0
y
x a
2
and you want to solve for x and y. First create the necessary
symbolic objects.
There are several ways to address the output of solve.
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Several Algebraic Equations
One is to use a two-output call
which returns
x y 0
y
x a
2
2
2
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Exercise#3
1. Solve following system of linear equations.
3x  4 y  4
x  9y  3
2. Solve following system of linear equations.
4 x  3 y  9 z  11
2 x  2 x  3z  5
x  5y  z  6
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Symbolic Matrix Operations
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Symbolic Matrix
This toolbox lets you represent matrices in symbolic form as well as MATLAB’s
numeric form. Given the numeric matrix:
The function sym(a) converts a to the symbolic matrix. The result is
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Symbolic Matrix
You can Also create a symbolic matrix in the following manner
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Determinant of Symbolic Matrix
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Diagonal of Symbolic Matrix
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Eigen Values of Symbolic Matrix
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Inverse of Symbolic Matrix
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Jordan Canonical of Symbolic Matrix
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Characteristic Polynomial of Symbolic Matrix
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Rank of Symbolic Matrix
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Lower Triangle of Symbolic Matrix
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Upper Triangle of Symbolic Matrix
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Exercise#4
1. Find determinant, inverse, rank, polynomial and eigen values
of the following symbolic matrix.
 x
A   x  y
 y  z
x y
y
xz
x  y  z
y  z 

z
2. Convert symbolic matrix A to Jordan, upper triangle and lower
triangle form.
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Function Calculator
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Function Calculator
• funtool is a visual function calculator that manipulates and displays
functions of one variable.
• At the click of a button, for example, funtool draws a graph
representing the sum, product, difference, or ratio of two functions that
you specify.
• funtool includes a function memory that allows you to store functions
for later retrieval.
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Function Calculator
• At startup, funtool displays graphs of a pair of functions, f(x) = x and
g(x) = 1.
• The graphs plot the functions over the domain [-2π, 2 π].
• funtool also displays a control panel that lets you save, retrieve,
redefine, combine, and transform f and g.
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Function Calculator
When you will type funtool in the command following GUI will appear
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Exercise#05
Solve the following functions using funtool.
(a)
f  3t  t  1 (b) g  t  1
2
(c) num g
3
(d)

fdt
3
(g) f(g)
df
dt
(f) den g
(h) f(x+3)
(g) g(x+1)
(e)
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Taylor Series Calculator
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Taylor Series Calculator
• taylortool initiates a GUI that graphs a function against the Nth
partial sum of its Taylor series about a base point x = a.
• The default function, value of N, basepoint, and interval of
computation for taylortool are f = x*cos(x),N = 7, a = 0, and
[-2*pi,2*pi], respectively.
• taylortool('f') initiates the GUI for the given expression f
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Taylor Series Calculator
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Exercise#06
1. Represent following functions in Taylor series of fourth
order using taylortool.
f e
sin( x )
2. Represent following functions in Taylor series of 10th
order using taylortool.
f  ln(1  x)
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Ordinary Differential Equations(ODEs)
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Ordinary Differential Equations
In MATLAB we use D to denotes differential in an ordinary
differential equation. For example
y'3 y  2
can be entered in MATLAB as
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Ordinary Differential Equations
The symbols D2, D3, ... DN, correspond to the second, third, ..., Nth
derivative, respectively. The Differential equation
y' ' '  2 y  5
can be entered in MATLAB as
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Ordinary Differential Equations
Consider following ODE
y' '  cos(2 x)  y
can be entered in MATLAB as
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Solution of Ordinary Differential Equation
• The function dsolve computes symbolic solutions to ordinary differential
equations.
• The dependent variables are those preceded by D and the default
independent variable is t.
Consider this equation
y'3 y  2
dy
 3y  2
dt
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Solution of Ordinary Differential Equation
• To change the default variable
y'3 y  2
dy
 3y  2
dx
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Solution of ODE (with initial conditions)
• To specify an initial condition, use the second input argument as
y'3 y  2,
y(0)  3
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Solution of non-linear ODE (with initial conditions)
• Nonlinear equations may have multiple solutions, even when initial
conditions are given:
( y' )  x  1,
2
2
x(0)  0
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Solution of Higher Order ODE (with initial conditions)
• Here is a second order differential equation with two initial conditions.
The commands
2
d y
dy
 cos(2 x)  ,
2
dx
dx
y (0)  0,
y ' (0)  0
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Solution of Higher Order ODE (with initial conditions)
• Here is a third order differential equation with three initial conditions.
3
d u
 u,
3
dx
u (0)  1, u ' (0)  1, u ' ' (0)  
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Exercise#7
Solve following differential equations
dy
 4 y  e t
dt
d2y
2 x

4
y

e
dx2
d2y
 xy, y(0)  0, y' (0)  1
2
dx
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Exercise#7 (contd….)
Solve following differential equations
dy
 4 y  yz ,
dz
y (3)  1
d2y
2 x
 4 y  e , y(0)  1, y(2)  0, y' (1)  8
2
dx
d3y
 xy, y(0)  1, y' (1)  1, y' ' (4)  1
3
dx
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Solution of Several ODE
• The function dsolve can also handle several ordinary differential
equations in several variables, with or without initial conditions.
• For example, here is a pair of linear, first-order equations.
df
 3 f  4g
dt
dg
 4 f  3g
dt
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Solution of Several ODE (with initial Conditions)
• If you prefer to recover f and g with initial conditions, type
df
 3 f  4g
dt
dg
 4 f  3 g
dt
f ( 0)  1
g ( 0)  2
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Exercise#8
Solve following pairs of differential equations
dy
 4 y  yz
dt
dz
 4 zy
dt
dy
 xy,
dt
dx
 4 x  y,
dt
y ( 0)  1
x ( 0)  0
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Thank you for your Attention
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