Power series solution of differential equations calculator wolfram

Q: Can Wolfram Alpha solve system of differential equations?

The Wolfram Languages differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without needing preprocessing by the user.

Table of Contents Show

Number Line
Details and Options
Basic Examples (2)
Scope (3)
Options (1)
Applications (2)
Properties and Relations (2)
Possible Issues (2)
Related Links
Version History
Related Resources
How do you solve a differential equation with a power series?
Can Wolfram Alpha solve system of differential equations?
Why we use power series solution of differential equations?
What is meant by power series solution?

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Examples

separable\:y'=e^{-y}(2x-4)
separable\:\frac{dr}{d\theta}=\frac{r^2}{\theta}
separable\:y'=\frac{xy^3}{\sqrt{1+x^2}}
separable\:y'=\frac{xy^3}{\sqrt{1+x^2}},\:y(0)=-1
separable\:y'=\frac{3x^2+4x-4}{2y-4},\:y(1)=3

separable-differential-equation-calculator

Compute the formula for the Frobenius series solution to an ODE

Contributed by: Rauan Kaldybayev

ResourceFunction["FrobeniusDSolveFormula"][eqn,u,x,x0]

outputs a formula to be fed into the resource function FrobeniusDSolve.

ResourceFunction["FrobeniusDSolveFormula"][eqn,u,x,x0,Association]

outputs the formula for the Frobenius series solution

of a differential equation for the function u, with independent variable x.

Details and Options

The output is a formula describing a Frobenius series

. It is a list of the form {elem1,elem2,…}, where the elemi gives information about the ith solution.

Solving ODEs using Frobenius's method is a tedious monotonous work, yet it is very simple - a perfect process to automate. ResourceFunction["FrobeniusDSolveFormula"] can be used to compute a formula for the Frobenius series solution to an ODE

. Computing a Frobenius series solution by hand can lead to potential insights or be helpful in conceptually understanding the behavior of the solutions.

Another application for ResourceFunction["FrobeniusDSolveFormula"] is to assist the resource function FrobeniusDSolve. When solving an ODE, FrobeniusDSolve first utilizes ResourceFunction["FrobeniusDSolveFormula"] to compute a formula for the solution. The formula is then used to calculate the approximate solutions. When a single ODE is to be solved many times with different parameters, computation time can be significantly reduced by computing the formula separately, using ResourceFunction["FrobeniusDSolveFormula"]. (see Applications)

The function can be used to solve linear homogeneous ODEs with polynomial coefficients.

Any linear ODE with polynomial coefficients can be written in the form

for some (any) x0. Frobenius's method assumes solution of the form

; substituting this into the equation yields

, where

. To satisfy the equation, the coefficients before all powers of x-x0 must be zero for all k. From here, we can determine the possible values of the constant r for which a0≠0, and then for each r, compute the coefficients. Read more at the related Wolfram Community post.

The options for ResourceFunction["FrobeniusDSolveFormula"] are:

"ComputeError"

False

whether to compute the error of the approximate solution

Examples

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Basic Examples (2)

Obtain a formula for the Frobenius solution for the ODE 2x2y''+7x(x+1)y'-3y=0 near the singular point x=0:

In[1]:=

Out[1]=

The formula describes a Frobenius series

. Display the same formula in a form compatible with the resource function FrobeniusDSolve (see Properties and Relations):

In[2]:=

Out[2]=

Scope (3)

The function can operate with complex numbers:

In[3]:=

Out[3]=

The inputs of the function can be symbolics:

In[4]:=

Out[4]=

FrobeniusDSolveFormula can be used to solve ODEs of any order. Here is the formula for the Frobenius series solution to (x-1)4f''''[x]=x(2+x)f[x]+x2(x-1)2f''[x]-19(x-1)3f'''[x] near the singular point x=1:

In[5]:=

Out[5]=

Options (1)

If the option "ComputeError" is set to True, FrobeniusDSolveFormula will not only output a formula for the Frobenius series solution but also the formula for the error of this solution:

In[6]:=

Out[6]=

The formula doesn't make much sense to a human and instead is meant to be fed into the resource function FrobeniusDSolve, which will in turn compute an approximate solution of the form

and the error of this solution.

Applications (2)

FrobeniusDSolve is a related resource function that computes approximate Frobenius and power series solutions to ODEs. The solutions it outputs are of the form

for some parameters x0, j. The following is an example of how FrobeniusDSolve can be used to obtain approximate solutions to the ODE x2y''-2xy'+(1+x)y=0 with x0=0 and n=4:

In[7]:=

Out[7]=

When solving an ODE, FrobeniusDSolve first utilizes FrobeniusDSolveFormula to compute a formula for the solution. The formula is then used to calculate the approximate solutions. When a single ODE is to be solved many times with different parameters, computation time can be significantly reduced by computing the formula separately, using FrobeniusDSolveFormula. This way, FrobeniusDSolve doesn't have to compute the formula every time, and it only has to do the numerical part of the computation. As can be seen from the following example, the decrease in computation time is dramatic:

In[8]:=

Out[9]=

Properties and Relations (2)

Formulas given by FrobeniusDSolveFormula can be fed into the resource function FrobeniusDSolve to compute approximate Frobenius series solutions. This can help significantly improve efficiency when a single ODE is to be solved repeatedly (see Applications). Obtain a formula for the Frobenius solution for the ODE 2x2y''+7x(x+1)y'-3y=0 near the singular point x=0, and output it in a form compatible with FrobeniusDSolve:

In[10]:=

Out[10]=

Use the formula to compute two linearly independent approximate solutions

In[11]:=

Out[11]=

In[12]:=

Out[12]=

Possible Issues (2)

The ODE must be linear and homogeneous, and its coefficients must be polynomial. Otherwise, FrobeniusDSolveFormula returns $Failed:

In[13]:=

Out[13]=

In[14]:=

Out[14]=

In[15]:=

Out[15]=

If the ODE isn't liner but can be made so by dividing by a non-zero factor, the function still doesn't accept it:

In[16]:=

Out[16]=

Publisher

Wolfram Summer Camp

http://www.its.caltech.edu/~esp/acm95b/frobenius.pdf

Version History

1.0.0 – 13 April 2021

How do you solve a differential equation with a power series?

Problem-Solving Strategy: Finding Power Series Solutions to Differential Equations.

Assume the differential equation has a solution of the form y(x)=∞∑n=0anxn..

Differentiate the power series term by term to get y′(x)=∞∑n=1nanxn−1. ... .

Substitute the power series expressions into the differential equation..

Can Wolfram Alpha solve system of differential equations?

The Wolfram Language's differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without needing preprocessing by the user.

Why we use power series solution of differential equations?

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

What is meant by power series solution?

Definition. A power series solution to a differential equation is a function with infinitely many terms, each term containing a different power of the dependent variable. The general solution has the form y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ {\displaystyle y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots } .