﻿ ﻿﻿﻿ math handbook calculator - Fractional Calculus Computer Algebra System software + + =  ﻿

# Examples of Fractional Calculus Computer Algebra System 例题

## Content

• Arithmetic 算术
• Algebra 代数
• Function 函数
• Calculus 微积分
• Equation 方程
• Discrete Math 离散数学
• Definition 定义式
• Numeric math 数值数学
• Number Theory 数论
• Probability 概率
• Multi elements
1. list()
2. vector
3. and
• Plot 制图
1. Interactive plot 互动制图
2. parametric plot, polar plot
3. solve equation graphically
4. dynamic plot 制图 with tangentplot
5. area plot with integral
6. complex plot
7. Geometry 几何
• plane graph 平面图
• 3D graph 立体图
1. surface in 3D with plot3D
2. contour in 3D with contour3D
3. wireframe in 3D with wirefram3D
4. complex function in 3D with complex3D
5. a line in 3D with parametric3D
6. a column in 3D with parametric3D
7. the 4-dimensional object (x,y,z,t) in 3D with implicit3D
• Animation 动画
• programming 编程
• bugs

## Arithmetic 算术 >>

### Exact computation

• Fraction 1E2-1/2
• Big number: add prefix "big" to number
big1234567890123456789

• mod operation
input mod(3,2) for 3 mod 2

### Complex 复数

Complex( 1,2) number is special vector, i.e. the 2-dimentional vector, so it can be operated and plotted as vector.
• complex numbers in the complex plane
complex(1,2) = 1+2i
• input complex number in polar(r,theta*degree) coordinates
polar(1,45degree)

• input complex number in polar(r,theta) coordinates for degree by polard(r,degree)
polard(1,45)

• input complex number in r*cis(theta*degree) format
2cis(45degree)
• #### Convert to complex( )

• in order to auto plot complex number as vector, input complex(1,-2) for 1-2i, or convert 1-2i to complex(1,-2) by
convert(1-2i to complex) = tocomplex(1-2i)
• input complex number in polar
tocomplex(polar(1,45degree))

• Convert complex a+b*i to polar(r,theta) coordinates
convert 1-i to polar = topolar(1-i)

• Convert complex a+b*i to polar(r,theta*degree) coordinates
topolard(1-i)

• plot complex

### Numerical approximations

• Convert back by numeric computation n( )
n(polar(2,45degree))
n( sin(pi/4) )
n( sin(30 degree) )
• sin^((0.5))(1) is the 0.5 order derivative of sin(x) at x=1
n( sin(0.5,1) )
• sin(1)^(0.5) is the 0.5 power of sin(x) at x=1
n( sin(1)^0.5 )
• ## Algebra 代数 >>

• simplify
taylor( (x^2 - 1)/(x-1) )
• expand
expand( (x-1)^3 )

• factorization
factor( x^4-1 )
• factorizing
factor( x^2+3*x+2 )
• ### tangent

• tangent equation at x=1
tangent( sin(x),x=1 )
• tangentplot( ) show dynamic tangent line when your mouse over the curve.
tangentplot( sin(x) )

### Convert

convert( sin(x) to exp) is the same as toexp(sin(x))
• convert to exp
toexp( cos(x) )

• convert to trig
convert exp(x) to trig

• convert sin(x) to exp(x),
convert sin(x) to exp = toexp( sin(x) )

• Convert to exp(x)
toexp(Gamma(2,x))
• ### inverse function

• input sin(x), click the inverse button
inverse( sin(x) )
check its result by clicking the inverse button again.
In order to show multi-value, use the inverse equation instead function.

### inverse equation

• inverse equation to show multivalue if it has.
inverse( sin(x)=y )
check its result by clicking the inverse button again.

### polynomial

• the unit polynomial
poly(3,x) gives the unit polynomial x^3+x^2+x+1.

• Hermite polynomial
hermite(3,x) gives the Hermite polynomial while hermite(3) gives Hermite number.

• harmonic polynomial
harmonic(-3,1,x) = harmonic(-3,x)
harmonic(-3,x)

• the zeta polynomial
zeta(-3,x) is the zeta polynomial.

• expand polynomial
expand(hermite(3,x))

• topoly( ) convert polynomial to polys( ) as holder of polynomial coefficients,
convert x^2-5*x+6 to poly = topoly( x^2-5*x+6 )
• activate polys( ) to polynomial
simplify( polys(1,-5,6,x) )
• polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
convert (x^2-1) to polyroot = topolyroot(x^2-1)
• polysolve( ) numerically solve polynomial for multi-roots.
polysolve(x^2-1)
• nsolve( ) numerically solve for a single root.
nsolve(x^2-1)
• solve( ) for sybmbloic and numeric roots.
solve(x^2-1)
• construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
simplify( polyroots(2,3) )

### Number

When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.
• ## Function 函数 >>

Function 函数
• Trigonometry 三角函数
expand Trigonometry by expandtrig( )

• inverse function
inverse( sin(x) )

• plot a multivalue function by the inverse equation
inverse( sin(x)=y )

• expand
expand( sin(x)^2 )
• factor
factor( sin(x)*cos(x) )

• ### Complex Function 复变函数

complex2D( ) shows the real and imag curves in real domain x, and complex3D( ) shows complex function in complex domain z, for 20 graphes in one plot.

## Calculus 微积分 >>

### Limit

• click the lim( ) button to Limit at x->0
lim_(x->0) sin(x)/x  = lim sin(x)/x as x->0 = lim(sin(x)/x)
• click the nlim( ) button to numeric limit at x->0
• click the limoo( ) button to Limit at x->oo
lim _(x->oo) log(x)/x = lim( log(x)/x as x->inf )

### Derivatives

• Differentiate
d/dx sin(x) = d(sin(x))

• Second order derivative
d^2/dx^2 sin(x) = d(sin(x),x,2) = d(sin(x) as x order 2)

• sin(0.5,x) is inert holder of the 0.5 order derivative sin^((0.5))(x), it can be activated by simplify( ):
simplify( sin(0.5,x) )
• Derivative as x=1
d/dx | _(x->1) x^6 = d( x^6 as x->1 )

• Second order derivative as x=1
d^2/dx^2| _(x->1) x^6 = d(x^6 as x->1 order 2) = d(x^6, x->1, 2)

#### Fractional calculus

• Fractional calculus
• semiderivative
d^(0.5)/dx^(0.5) sin(x) = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x))

• input sin(0.5,x) as the 0.5 order derivative of sin(x) for
sin^((0.5))(x) = sin^((0.5))(x) = sin(0.5,x)
• simplify sin(0.5,x) as the 0.5 order derivative of sin(x),
sin^((0.5))(x) = simplify(sin(0.5,x))
• 0.5 order derivative again
d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x) = d(d(sin(x),x,0.5),x,0.5)
• Minus order derivative
d^(-0.5)/dx^(-0.5) sin(x) = d(sin(x),x,-0.5)

• inverse the 0.5 order derivative of sin(x) function
f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x))

• Derive the product rule
d/dx (f(x)*g(x)*h(x)) = d(f(x)*g(x)*h(x))

• … as well as the quotient rule
d/dx f(x)/g(x) = d(f(x)/g(x))

• for derivatives
d/dx ((sin(x)* x^2)/(1 + tan(cot(x)))) = d((sin(x)* x^2)/(1 + tan(cot(x))))

• Multiple ways to derive functions
d/dy cot(x*y) = d(cot(x*y) ,y)

• Implicit derivatives, too
d/dx (y(x)^2 - 5*sin(x)) = d(y(x)^2 - 5*sin(x))

• the nth derivative formula
 d^n/dx^n (sin(x)*exp(x))  = nthd(sin(x)*exp(x))
• #### differentiate graphically

some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in plot2D. e.g.

### Integrals

• click the ∫ button to integrate above result
int(cos(x)*e^x+sin(x)*e^x)\ dx = int(cos(x)*e^x+sin(x)*e^x)
int tan(x)\ dx = integrate tan(x) = int(tan(x))

• Multiple integrate
int int (x + y)\ dx dy = int( int(x+y, x),y)
int int exp(-x)\ dx dx = integrate(exp(-x) as x order 2)

• Definite integration
int _1^3 (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

• Improper integral
int _0^(pi/2) tan(x) dx =int(tan(x),x,0,pi/2)

• Infinite integral
int _0^oo 1/(x^2 + 1) dx = int(1/x^2+1),x,0,oo)

• indefinite integrate int sin(x) dx = integrate(sin(x))

• Definite integration
int_0^1 sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1

#### fractional integrate

• semi integrate, semiint( )
int sin(x) \ dx^(1/2) = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2)

• indefinite semiintegrate
int sin(x)\ dx^0.5 = d^(-0.5)/dx^(-0.5) sin(x) = int(sin(x),x,0.5) = semiint(sin(x))

• Definite fractional integration
int_0^1 sin(x) (dx)^0.5 = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1

int (2x+3)^7 dx = int (2x+3)^7

• numeric computation by click on the "~=" button
n( int _0^1 sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))

#### integrate graphically

some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in plot2D. e.g.
• ## Equation 方程 >>

### inverse an equation

• inverse an equation to show multivalue curve.
inverse( sin(x)=y )
check its result by clicking the inverse button again.

### polynomial equation

• polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
convert (x^2-1) to polyroot = topolyroot(x^2-1)

• polysolve( ) numerically solve polynomial for multi-roots.
polysolve(x^2-1)

• construct polynomial from roots. activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
simplify( polyroots(2,3) )

• solve( ) for sybmbloic and numeric roots.
solve(x^2-1)
solve( x^2-5*x-6 )

• solve equation and inequalities, by default, equation = 0 for default unknown x if the unknown omit.
solve( x^2+3*x+2 )

• Symbolic roots
solve( x^2 + 4*x + a )

• Complex roots
solve( x^2 + 4*x + 181 )

• solve equation for x.
solve( x^2-5*x-6=0,x )

• numerically root
nsolve( x^3 + 4*x + 181 )

• nsolve( ) numerically solve for a single root.
nsolve(x^2-1)

### Algebra Equation

solve( ) also solve other algebra equation, e.g. exp( ) equation,
• Solve nonlinear equations:
solve(exp(x)+exp(-x)=4)

### system of equations

• system of 2 equations with 2 unknowns x and y by default if the unknowns omit.
solve( 2x+3y-1=0,x+y-1=0 )

### Diophantine equation

• it is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
solve( 3x-2y-2=0, x,y)

### congruence equation

By definition of congruence, a x ≡ b (mod m) if a x − b is divisible by m. Hence, a x ≡ b (mod m) if a x − b = m y, for some integer y. Rearranging the equation to the equivalent form of Diophantine equation a x − m y = b.
3x-2=2*(mod 2)
3x-2=2mod(2)

### Modulus equation

• solve( ) Modulus equation for the unknown x inside the mod( ) function, e.g.
input mod(x,2)=1 for
x mod 2 = 1
click the solve button

### Probability_equation

• solve( ) Probability equation for the unknown k inside the Probability function P( ),
solve( P(x>k)=0.2, k)

### recurrence_equation

• rsolve( ) recurrence equation for f(x)
f(x+1)=f(x)+x
f(x+1)=f(x)+1/x

### functional_equation

• rsolve( ) functional equation for f(x)
f(x-y)=f(x)/f(y)
f(x*y)=f(x)+f(y)

### Inequalities

• solve( ) Inequalities for x.
solve( 2*x-1>0 )
solve( x^2+3*x+2>0 )

### differential equation

ODE( ) and dsolve( ) and lasove( ) solve ordinary differential equation (ODE) to unknown y.
• Solve linear ordinary differential equations:
y'=x*y+x
y'= 2y
y'-y-1=0

• Solve nonlinear ordinary differential equations:
(y')^2-2y^2-4y-2=0
• 2000 examples of Ordinary differential equation (ODE)

#### solve graphically

The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the checkbox. by default it is first order ODE.
y''=y'-y for second order ODE

### integral equation

#### indefinite integral equation

• indefinite integral equation
input ints(y) -2y = exp(x) for
int y dx - 2y = exp(x)

#### definite integral equation

• definite integral equation
input integrates(y(t)/sqrt(x-t),t,0,x) = 2y for
int_0^x (y(t))/sqrt(x-t) dt = 2y

### differential integral equation

input ds(y)-ints(y) -y-exp(x)=0 for
dy/dx-int y dx -y-exp(x)=0

### fractional differential equation

dsolve( ) also solves fractional differential equation
• Solve linear equations:
d^0.5/dx^0.5 y = 2y
d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0
d^0.5/dx^0.5 y -y -exp(4x) = 0
(d^0.5y)/dx^0.5=sin(x)

• Solve nonlinear equations:
d^0.5/dx^0.5 y = y^2

#### fractional integral equation

• d^-0.5/dx^-0.5 y = 2y

#### fractional differential integral equation

ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0
(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0

#### complex order differential equation

• (d^(1-i) y)/dx^(1-i)-2y-exp(x)=0

#### variable order differential equation

(d^sin(x) y)/dx^sin(x)-2y-exp(x)=0

### system of differential equations

• system of 2 equations with 2 unknowns x of the 0.5 order and y of the 0.8 order with a variable t.
dsolve( x(1,t)=x,y(1,t)=x-y )

### partial differental equation

PDE( ) and pdsolve( ) solve partial differental equation with two variables t and x, then click the plot2D button to plot solution, pull the t slider to change the t value. click the plot3D button for 3D graph.
• Solve a linear equation:
dy/dt = dy/dx-2y

• Solve a nonlinear equation:
dy/dt = dy/dx*y^2

#### fractional partial differental equation

PDE( ) and pdsolve( ) solve fractional partial differental equation.
• Solve linear equations:
(d^0.5y)/dt^0.5 = dy/dx-2y

• Solve nonlinear equations:
(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5*y^2

• More examples are in Analytical Solution of Fractional Differential Equations

### test solution

#### test solution for algebaic equation to the unknown x by test(solution,eq,x) or click the test( ) button.

• test(1,x^2-1=0,x)
test( -1, x^2-5*x-6 )

#### test solution for differential equation to the unknown y by test( ) or click the test( ) button.

• test( exp(2x), dy/dx=2y )
• test( exp(4x), (d^0.5y)/dx^0.5=2y )
• ## Discrete Math 离散数学 >>

The default index variable in discrete math is k.
• Input harmonic(2,x), click the defintion( ) button to show its defintion, check its result by clicking the simplify( ) button, then click the limoo( ) button for its limit as x->oo.

### Difference

Δ(k^2) = difference(k^2)
• Check its result by the sum( ) button

### Summation ∑

#### Indefinite sum

∑ k = sum(k)
• Check its result by the difference( ) button
Δ sum(k) = difference( sum(k) )
• In order to auto plot, the index variable should be x,
sum_x x = sum(x,x)

#### definite sum

• Definite sum = Partial sum x from 1 to x, e.g.
1+2+ .. +x = sum _(k=1) ^x k = sum(k,k,1,x)
• Definite sum, sum x from 1 to 5, e.g.
1+2+ .. +5 = ∑(x,x,0,5) = sum(x,x,0,5)
• Infinite sum x from 0 to inf, e.g.
1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo
sum(x^k,k,0,5)

• #### Definite sum with parameter x as upper limit

sum(k^2, k,0, x)
• Check its result by the difference( ) button, and then the expand( ) button.
• convert to sum series definition
tosum( exp(x) )
• expand above sum series by the expand( ) button
expand( tosum(exp(x)) )

• #### Indefinite sum

∑ k
sum( x^k/k!,k )
• partial sum of 1+2+ .. + k = ∑ k = partialsum(k)
• Definite sum of 1+2+ .. +5 = ∑ k

#### partial sum with parameter upper limit x

sum(1/k^2,k,1,x)

#### infinite sum

• if the upper limit go to infinite, it becomes infinite sum
Infinite sum of 1/1^2+1/2^+1/3^2 .. +1/k^2+... = lim(sum( 1/k^2,k,1,x) as x->oo) = sum( 1/k^2,k,1,oo )
Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = lim(sum( 1/k!,k,0,x) as x->oo) = sum( 1/k!,k,0,oo )
• ### Series 级数

• convert to sum series definition
tosum( exp(x) ) = toseries( exp(x) )
• check its result by clicking the simplify( ) button
simplify( tosum( exp(x) ))

• expand above sum series
expand( tosum(exp(x)) )

• compare to Taylor series
taylor( exp(x), x=0, 8)
• compare to series
series( exp(x) )

• Taylor series expansion as x=0,
taylor( exp(x) as x=0 ) = taylor(exp(x))

by default x=0.

• series expand not only to taylor series,
series( exp(x) )
but aslo to other series expansion,
series( zeta(2,x) )

### Product ∏

• prod(x,x)

• prod x

## Definition 定义式 >>

• definition of function
definition( exp(x) )
• check its result by clicking the simplify( ) button
simplify( def(exp(x)) )
• #### series definition

• convert to series definition
toseries( exp(x) )
• check its result by clicking the simplify( ) button
simplify( tosum(exp(x)) )
• #### integral definition

• convert to integral definition
toint( exp(x) )
• check its result by clicking the simplify( ) button
simplify( toint(exp(x)) )
• ## Numeric math 数值数学 >>

• numeric solve equation,
nsolve( x^2-5*x+6=0 )
nsolve( x^2-5*x+6 )

• numeric integrate, by default x from 0 to 1.
nint( x^2-5*x+6,x,0,1 )
nint x^2-5*x+6 as x from 0 to 1
nint sin(x)
• numeric computation,
n( sin(30 degree) ) = n sin(30 degree)

• ## Number Theory 数论 >>

When the variable x of polynomial is numnber, it becomes number, e.g.
• poly number
poly(3,2)
• Hermite number
hermite(3,2)
• harmonic number
harmonic(-3,2)
harmonic(-3,2,4)
harmonic(1,1,4) = harmonic(1,4) = harmonic(4)
• Bell number
n(bell(5))
• double factorial 6!!
• Calculate the 4nd prime prime(4)
• is prime number? isprime(12321)
• next prime greater than 4 nextprime(4)
• binomial number ((4),(2))
• combination number C_2^4
• harmonic number H_4
• congruence equation
3x-1=2*(mod 2)
3x-1=2mod( 2)
• modular equation
Enter mod(x-1,10)=2 for (x-1) mod 10=2
• Diophantine equation
number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,
solve( 3x-2y-2=0, x,y )
• ## Probability 概率 >>

• P( ) is probability of standard normal distribution
P(x<0.8)
• Phi( ) is standard normal distribution function
Phi(x)
• solve Probability equation for k
solve(P(x>k)=0.2,k)
• ## Multi elements >>

We can put multi elements together with list(), vector(), and. Most operation in them is the same as in one element, one by one. e.g. +,-,*,/, differentiation, integration, sum, etc. We count its elements with size(), as same as to count elements in function. We only talk about special properties as follows.
1. list()
We can sort list(), add numbers together with total(list()), max(list()), min(list()).
2. ### vector( )

It has direction. the position of the element is fix so we cannot sort it. vector is number with direction. two vector( ) in the same dimention can be operated by +, -, *, /, ^. the result can be checked by its reverse operation. the system auto plot the 2-dimentional vector.
vector(2,4)/vector(1,2) =2
vector(2,4)=vector(1,2)*2
3. and
We can plot multi curves with the and. e.g. plot(x and x*x). the position of the element is fix so we cannot sort it.

## Plot 制图 >>

#### Interactive plot 互动制图

• plot 制图 The system auto plot when you hit the ENTER key, move mouse wheel on graph to zoom and move.

#### solve equation graphically

• input sin(x)-0.5=0, hit the ENTER key or click the plot button, then put mouse on cross between its curve and the x-axis to show the x-axis value for solution ,
sin(x)-0.5=0
• plot sin(x) and x^2, put mouse on cross between two curves to show solution.
plot( sin(x) and x^2)
• #### implicit plot

• implicit plot sin(x)=y to show a multivalue function,
implicitplot( x=sin(y) )
• #### parametric plot

• parametric plot with default pararmter t
parametricplot( sin(t) and sin(4*t) )
• #### polar plot

polarplot( 2*sin(4*x) )

#### dynamic plot

• tangent plot, by moving mouse on the curve to show tangent
tangentplot( sin(x) )
• secant plot, by moving mouse on the curve to show secant
secantplot( sin(x) )
• #### overlap plot

• overlap plot by clicking the overlap button or with overlapplot

#### complex plot

plot complex number with tocomplex( ), or enter complex number in complex(1,1) format, e.g.
tocmplex(1-i)

plot complex function in real domain with complex2D( ), e.g.
complex2D(sqrt(x))

plot complex function in complex domain with complex3D( ), e.g.
complex3D(sqrt(x)) for 20 graphes in one plot.

More examples in complex function

### area plot with integral

• integral( ) plot the area under integral curve.
• ### Geometry 几何

• 2 semicircle with radius 1, 半园
-semicircle(1) and semicircle(1)
• integral area by integral( )
integral(semicircle(2))

• circle with radius 2, 园
circle(2)
oval(2,1)
• tangent 切线 at x=0 by default,
tangent( sin(x) as x=1 )

#### Dynamic tangent 切线

tangentplot( ), then put mouse over curve to show a tangent line and equation,
tangentplot( sin(x) )

#### Dynamic secant 割线

• secant at x=0 by default, secantplot( ), then put mouse over curve to show a secant,
secantplot( sin(x) )
• more examples on graph

## plane graph 平面图 plot 2D >>

### plot2D

plot2D a curve, derivative and integral curve in 2D and polar, and calculate expression at any x and y=0.
• plot2D(sin(x))
More examples in plane curve in 2D

#### the 3-dimensional object (x,y,t) in 2D plane

• plot2D the 3-dimensional object (x,y,t) in 2D plane, manually change the t value by a slider, or tick the auto t checkbox for auto change.
plot2D(x*y-t-1)

### function plot with funplot

• funplot a curve, inverse curve in 2D, and calculate expression at any x and y=0.

### differentiate graphically with diff2D

• diff2D numericallyly and graphically differentiate a function on graph.
diff2D(sin(x)) first order derivative

### semidifferentiate graphically with semid2D

• semid2D numerically and graphically semidifferentiate a function on graph.
semid2D(sin(x)) half order derivative graph
d(x=>sin(x),x,0.5) half order derivative

### differentiate graphically with d(x=>sin(x))

d(x=>sin(x)) first order derivative graph
d(x=>sin(x),x,2) second order derivative
d(x=>sin(x),x,0.5) half order derivative

### integrate graphically with integrate2D

• integrate2D numerically and graphically integrate a function on graph. it convert the integrate( ) to the integrates(x=>sin(x)) for integral graph.

### solve ODE graphically with odeplot

• odeplot graphically solve Ordinary Differential Equation (ode). The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the y'= checkbox for first order differential_equation, or click the y"= checkbox for second order differential_equation, by default it is first order ODE.
y''=y'-y for second order ODE

• ## 3D graph 立体图 plot 3D >>

All 3D graphes are interactive, zoom by mouse wheel, and animation by clicking the spin checkbox.
• with plot3D

#### plot3D plot with 1 parameter and 1,2,3 variables

• surface in 3D space by Internet Brower plot3DIE(sin(x))
• surface in 3D with plot3D(sin(x))
• contour in 3D with contour3D(x*y)
• wireframe in 3D with wireframe3D(x*y)
• a line in 3D space with parametric3D(x*x)
More examples in space curve in 3D
• a column in 3D space with parametric3D(x*x+y*y-1)
• implicit3D graphically solve the 3-dimensional equation
implicit3D(x*y*z-1=0)
More examples in Surface in 3D

• complex function in 3D with complex3D(sqrt(x)) for 20 graphes in one plot.
More examples in complex function

#### the 4-dimensional object (x,y,z,t) in 3D space

• the 4-dimensional object (x,y,z,t) in 3D space, manually change the t value by a slider
implicit3D(x*y*z-t)
More examples in Quadratic hypersurfaces in 4D

#### Plot 3D with 3 parameters and 1 variable t or x

• parametric3D(cos(t),sin(t),t)
• parametric3D(sin(x),cos(x),x)

#### Plot 3D with 3 parameters and 2 variables x and y

• parametric3Dxy(x,y,x*y)
• wireframe3Dxy(x,y,x*y)
more examples

## programming 编程 >>

online programming 在线编程
1. plot 函数图
2. rose 玫瑰花
3. check code 验证码
4. calculator 计算器
5. sci calculator 科学计算器
6. color 颜色取色器
7. Chinese calendar 农历日历
8. calendar 日历

﻿