Equation is a statement of an equality containing one or more variables. You can create PNG images with equations using the Software:LaTeX syntax using the class  jhplot.HLatexEq. These images can be included on the web page or presentations. Here is a small code that shows how to make a PNG image

from jhplot import *
eq="\int^{100}_{i=0} F(x) dx" # use LaTeX syntax for this equation
image="/tmp/equation.png"     # output file with PNG image of this equation
q1=HLatexEq(eq, 32)           # create PNG image from LaTeX using the font size 32 
q1.export(image)              # making the image
print "Created :",image
IView(image)                  #  View the created PNG image

Numerous Java packages can be used to solve linear, non-linear and differential equations.

To solve linear, quadratic and cubic equations, use the  jhplot.math.Numeric package. In general, only real solutions are considered.

from jhplot.math.Numeric import *
a=solveLinear(1,2) # solves ax+b=0. a=1, b=2
print a
b= solveQuadratic(1, 2, -1) # roots of the quadratic equation
print b
b= solveCubic(1, 2, 4,2)
print b
c=solveQuartic(1,-2, 3, 4, -2)
print b

For solving polynomial equations, one can use the symbolic calculation engine.

from jhplot.math import *
from jhplot import *
j=Symbolic("jscl") # using jscl engine
j.expand("solve(c+b*x+a*x^2,x)") # answer: root[0](c, b, a)

One solve algebraic equation systems of any degree, with several indeterminates, by computing the Groebner bases of polynomial ideals. For example, let this system for the indeterminates x, y:

x^2 + y^2 = 4
x*y = 1

We use:

j.expand("groebner({x^2 + y^2 - 4, x*y - 1}, {x, y})")

The returned output is:

{1-4*x^2+x^4, 4*x-x^3-y}

which allows to find x, then y from x. This operation doesn't calculate the roots, it just writes the equation. For example, it wouldn't give "a = 4/5" but "5*a-4" ("= 0" implied). Groebner basis computation is explained in more details below:

Consider a linear systems of equations of the form AX=B. For example, consider

2x + 3y - 2z =  1
x + 7y + 6x = -2
4x - 3y - 5z =  1

We will solve this using  org.apache.commons.math3.linear.DecompositionSolver package:

from org.apache.commons.math3.linear import *

# get the coefficient matrix A using LU decomposition
coeff= Array2DRowRealMatrix([[2,3,-2],[-1,7,6],[4,-3,-5]])
solver =LUDecompositionImpl(coeff).getSolver()
#  use solve(RealVector) to solve the system 
constants = ArrayRealVector([1, -2, 1 ])
solution = solver.solve(constants);
print "Solution: x=",solution.getEntry(0), "y=",solution.getEntry(1),"z=",solution.getEntry(2)

The execution of this code prints:

Solution: x= -0.369863013699 y= 0.178082191781 z= -0.602739726027

Read more for different types of decomposition here. Please read more in the Linear Algebra section.

You can solve non-linear equations using the jMathLab symbolic kernel as explained in Section JMathLab Equations.

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