SymPy - Solvers

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Since the symbols = and == are defined as assignment and equality operators in Python, they cannot be used to formulate symbolic equations. SymPy provides Eq() function to set up an equation.

>>> from sympy import * 
>>> x,y=symbols('x y') 
>>> Eq(x,y)

The above code snippet gives an output equivalent to the below expression −

x = y

Since x=y is possible if and only if x-y=0, above equation can be written as −

>>> Eq(x-y,0)

The above code snippet gives an output equivalent to the below expression −

x − y = 0

The solver module in SymPy provides soveset() function whose prototype is as follows −

solveset(equation, variable, domain)

The domain is by default S.Complexes. Using solveset() function, we can solve an algebraic equation as follows −

>>> solveset(Eq(x**2-9,0), x)

The following output is obtained −

{−3, 3}

>>> solveset(Eq(x**2-3*x, -2),x)

The following output is obtained after executing the above code snippet −

{1,2}

The output of solveset is a FiniteSet of the solutions. If there are no solutions, an EmptySet is returned

>>> solveset(exp(x),x)

The following output is obtained after executing the above code snippet −

$\varnothing$

Linear equation

We have to use linsolve() function to solve linear equations.

For example, the equations are as follows −

x-y=4

x+y=1

>>> from sympy import * 
>>> x,y=symbols('x y') 
>>> linsolve([Eq(x-y,4),Eq( x + y ,1) ], (x, y))

The following output is obtained after executing the above code snippet −

$\lbrace(\frac{5}{2},-\frac{3}{2})\rbrace$

The linsolve() function can also solve linear equations expressed in matrix form.

>>> a,b=symbols('a b') 
>>> a=Matrix([[1,-1],[1,1]]) 
>>> b=Matrix([4,1]) 
>>> linsolve([a,b], (x,y))

We get the following output if we execute the above code snippet −

$\lbrace(\frac{5}{2},-\frac{3}{2})\rbrace$

Non-linear equation

For this purpose, we use nonlinsolve() function. Equations for this example −

a2+a=0 a-b=0

>>> a,b=symbols('a b') 
>>> nonlinsolve([a**2 + a, a - b], [a, b])

We get the following output if we execute the above code snippet −

$\lbrace(-1, -1),(0,0)\rbrace$

differential equation

First, create an undefined function by passing cls=Function to the symbols function. To solve differential equations, use dsolve.

>>> x=Symbol('x') 
>>> f=symbols('f', cls=Function) 
>>> f(x)

The following output is obtained after executing the above code snippet −

f(x)

Here f(x) is an unevaluated function. Its derivative is as follows −

>>> f(x).diff(x)

The above code snippet gives an output equivalent to the below expression −

$\frac{d}{dx}f(x)$

We first create Eq object corresponding to following differential equation

>>> eqn=Eq(f(x).diff(x)-f(x), sin(x)) 
>>> eqn

The above code snippet gives an output equivalent to the below expression −

$-f(x) + \frac{d}{dx}f(x)= \sin(x)$

>>> dsolve(eqn, f(x))

The above code snippet gives an output equivalent to the below expression −

$f(x)=(c^1-\frac{e^-xsin(x)}{2}-\frac{e^-xcos(x)}{2})e^x$

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