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AsymptoticDSolveValue
✖
AsymptoticDSolveValue

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✖
AsymptoticDSolveValue[eqn,f,xx0]

computes an asymptotic approximation to the differential equation eqn for f[x] centered at x0.

✖
AsymptoticDSolveValue[{eqn1,eqn2,…},{f1,f2,…},xx0]

computes an asymptotic approximation to a system of differential equations.

✖
AsymptoticDSolveValue[eqn,f,x,ϵϵ0]

computes an asymptotic approximation of f[x,ϵ] for the parameter ϵ centered at ϵ0.

✖
AsymptoticDSolveValue[eqn,f,…,{ξ,ξ0,n}]

computes the asymptotic approximation to order n.

Details and Options

  • Asymptotic approximations to differential equations are also known as asymptotic expansions, perturbation solutions, regular perturbations and singular perturbations, etc. They are also known by specific methods used to compute some of them, such as Frobenius series, WKB, boundary-layer method, etc.
  • Asymptotic approximations are typically used to solve problems for which no exact solution can be found or to get simpler answers for computation, comparison and interpretation.
  • AsymptoticDSolveValue[eqn,…,xx0] computes the leading term in an asymptotic expansion for eqn. Use SeriesTermGoal to specify more terms.
  • If the exact result is g[x] and the asymptotic approximation of order n at x0 is gn[x], then the result is AsymptoticLess[g[x]-gn[x],gn[x]-gn-1[x],xx0] or g[x]-gn[x]∈o[gn[x]-gn-1[x]] as xx0.
  • The asymptotic approximation gn[x] is often given as a sum gn[x]αkϕk[x], where {ϕ1[x],…,ϕn[x]} is an asymptotic scale ϕ1[x]≻ϕ2[x]≻⋯>ϕn[x] as xx0. Then the result is AsymptoticLess[g[x]-gn[x],ϕn[x],xx0] or g[x]-gn[x]∈o[ϕn[x]] as xx0.
  • Common asymptotic scales include:
  • Taylor scale when xx0
    Laurent scale when xx0
    Laurent scale when x±∞
    Puiseux scale when xx0
  • The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales.
  • The center x0 can be any finite or infinite real or complex number.
  • The order n must be a positive integer and specifies order of approximation for the asymptotic solution. It is not related to polynomial degree.
  • The specification u∈Vectors[n] or u∈Matrices[{m,n}] can be used to indicate that the dependent variable u is a vector-valued or a matrix-valued variable, respectively. Alternatively, u can be specified as a VectorSymbol or MatrixSymbol. » »
  • The following options can be given:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    GeneratedParameters Nonehow to name generated parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    SeriesTermGoalAutomaticnumber of terms in the approximation
    WorkingPrecisionAutomaticthe precision used in internal computations
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticDSolve typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Compute an asymptotic approximation for a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ei0ki0
Out[1]=1

Find a series solution for a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-g2tv8a
Out[1]=1

Plot the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bdyfy8
Out[2]=2

Find an asymptotic expansion for a perturbation problem:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-9vhgy
Out[1]=1

Plot the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gtg6xc
Out[2]=2

Scope  (49)Survey of the scope of standard use cases

Basic Uses  (8)

Compute a series solution of order 10 for an ODE around x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bvoaht
Out[1]=1

Plot the successive approximations for an asymptotic solution:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-drapx7
Out[1]=1

Use Accumulate to build the list of approximations:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-goabx1
Out[2]=2

Obtain the required plot:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-cu9qix
Out[3]=3

Compute a series solution around x=3:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b2gj32
Out[1]=1

Obtain a series approximation for the general solution:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-e51lj
Out[1]=1

Obtain series approximations with different numbers of terms:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-sp7km
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-f08uxt
Out[2]=2

Compute a series solution for a system of ODEs:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-cmfbs0
Out[1]=1

Compute a series solution for a perturbation problem:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-g5bnnd
Out[1]=1

Vary the number of terms:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gkgtav
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-l2fc5
Out[3]=3

Find an asymptotic solution for a perturbation problem:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-1ku72
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-er8n0v
Out[2]=2

Ordinary Points  (7)

Find a Taylor series solution for a linear first-order ODE at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-dfjlpw
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-y2va2
Out[2]=2

Series solution for a linear second-order ODE at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-rwe02
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fbprhh
Out[2]=2

Series solution for an inhomogeneous linear ODE at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fds63d
Out[1]=1

Series solution for a linear ODE with nonrational coefficients at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-darqy5
Out[1]=1

Series solution for a linear higher-order ODE at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b940gh
Out[1]=1

Series solution for a linear ODE at the ordinary point x=1:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-dzszzt
Out[1]=1

Series approximation for the general solution for a linear ODE at an ordinary point:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-hlxjv0
Out[1]=1

Regular Singular Points  (5)

Find a Frobenius series solution for a linear first-order ODE at the regular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bqpld7
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-puhegl
Out[2]=2

Series solution for a linear second-order ODE at the regular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-z19s0
Out[1]=1

Find a series solution for a linear higher-order ODE at the regular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bldpvi
Out[1]=1

The series solution is an exact solution in this case:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bmlicd
Out[2]=2

Series solution for a linear ODE at the regular singular point x=1:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-eeflq4
Out[1]=1

Series solution for a linear ODE with nonrational coefficients at the regular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ilgju5
Out[1]=1

Irregular Singular Points  (3)

Find an asymptotic solution for a linear first-order ODE at the irregular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-r7rdp
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-cje3fu
Out[2]=2

Series solution for a linear second-order ODE at the irregular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-d9cnp5
Out[1]=1

Series solution for a linear higher-order ODE at the irregular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-cz0dcs
Out[1]=1

Nonlinear ODEs  (7)

Find a series solution for a nonlinear first-order ODE at x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-lqw2p
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-czhbz5
Out[2]=2

Series solution for a nonlinear second-order ODE at x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-crrqp2
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bhi28a
Out[2]=2

Series solution for an inhomogeneous nonlinear ODE at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-exnc41
Out[1]=1

Series solution for a nonlinear ODE with nonrational coefficients at x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-c6qfk
Out[1]=1

Series solution for a nonlinear higher-order ODE at x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-t9ahg
Out[1]=1

Series solution for a nonlinear ODE at x=1:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-dj5ki8
Out[1]=1

Series approximation for the general solution of a nonlinear ODE:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-m12jdk
Out[1]=1

Solutions at Infinity  (4)

Find a series solution for a linear ODE at the ordinary point x=∞:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-nmge1r
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-hvgn8
Out[2]=2

Find a series solution for a linear ODE at the regular singular point x=∞:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-rudsg
Out[1]=1

Find a series solution for a linear ODE at the irregular singular point x=∞:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-lz3uxf
Out[1]=1

Find a series solution for a nonlinear ODE at x=∞:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-tmwju
Out[1]=1

Systems of ODEs  (7)

Find a series solution for a linear system of first-order ODEs at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fq63an
Out[1]=1

Plot the approximation given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bgpcwi
Out[2]=2

Series solution for a linear system of higher-order ODEs at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bcxrwt
Out[1]=1

Series solution for an inhomogeneous system of linear ODEs at the ordinary point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-dda9v
Out[1]=1

Series solution for a linear system of ODEs at the ordinary point x=1:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bx56u1
Out[1]=1

Series approximation to the general solution for a linear system of ODEs at an ordinary point:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-n3oj5r
Out[1]=1

Series solution for a linear system of ODEs at the ordinary point x=0 using vector variables:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-pwettv
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-imwalw
Out[2]=2

Alternatively, define as a VectorSymbol:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-cbnqix
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-l2ubf
Out[4]=4

Series solution for a linear system of ODEs at the ordinary point x=0 using matrix variables:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ceymi4
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-be9w1i

Alternatively, define as a MatrixSymbol:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-d3i8q
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-oxam09

Regular Perturbations  (2)

Find a series solution for a regular linear perturbation problem at ϵ=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bedaxw
Out[1]=1

Plot the successive approximations given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-eoxzwt
Out[2]=2

Find a series solution for a regular nonlinear perturbation problem at ϵ=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-eco6mt
Out[1]=1

Plot the approximation given by the solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-c0va6k
Out[2]=2

Singular Perturbations  (3)

Find a first-order approximation for a singular boundary value problem:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ux1ab
Out[1]=1

Plot the approximation for different values of the parameter:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-g0mrwi
Out[2]=2

Compare with a numerical solution:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-nkpwkv
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-dlgxi3
Out[4]=4

Find a second-order approximation for a singular boundary value problem:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gnlkap
Out[1]=1

Plot the approximation for different values of the parameter:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ewdf22
Out[2]=2

Find a second-order perturbation approximation at λ=∞:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fgo7a2
Out[1]=1

Plot the approximation for a large value of the parameter:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-i0bsj4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fpno8h
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fq9b3x
Out[4]=4

Fractional ODEs  (3)

Find the series solution for a linear fractional ODE of order 0.7:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ntmtv8
Out[1]=1

Solve the same ODE using DSolveValue:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-k7wdq0
Out[2]=2

Compare exact solution with asymptotic solutions for various values of approximation of order :

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-hjmd5z
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ej2zb1
Out[4]=4

Find the series solution for a linear fractional ODE with nonconstant coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-c4n43q
Out[1]=1

Find a series solution for a system of two linear fractional ODEs:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ptt6xv
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-mgvxwm
Out[2]=2

Plot the solution:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ji4dee
Out[3]=3

Options  (1)Common values & functionality for each option

GeneratedParameters  (1)

Use differently named constants:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-c14jm7
Out[1]=1

Use subscripted constants:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b12oz4
Out[2]=2

Applications  (7)Sample problems that can be solved with this function

Compute a Taylor polynomial approximation for Cos:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fpftr3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b9aiyg
Out[2]=2

Improve the range of the approximation by specifying a higher order:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-nx58qa
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ecxmqm
Out[4]=4

Study the variation of range with the order of the approximation:

In[5]:=5
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-l6lhkz
In[6]:=6
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-h1y0x1
Out[6]=6

Solve Bessel's equation of order around the regular singular point x=0:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bib12v
Out[1]=1

Plot the two components of the general solution:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-qb2sm
Out[2]=2

The Airy equation has an irregular singular point at x=∞:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-m96cvk

Compute an asymptotic expansion at the irregular singular point:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-nrt5cf
Out[2]=2

Compare with the expansions for the Airy functions at Infinity:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-upiyd
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b7x4yk
Out[4]=4

Plot the Airy functions and the approximations:

In[5]:=5
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gh3eac
Out[5]=5

Compute an exact polynomial solution of a nonlinear first-order ODE:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bmqjev
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-c8jqtk
Out[2]=2

Verify that this is a solution of the ODE:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-feaonb
Out[3]=3

Compute an equilibrium solution for a system of first-order ODEs:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-d8u0hl

Visualize the vector field defined by the system:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b3tmsw
Out[2]=2

Solve the system with general initial conditions:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-g7x5mw

Find the equilibrium solution:

In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gsokzw
Out[4]=4

Find a first-order perturbation expansion for the Duffing equation:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gsqnc1
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ghpwdr
Out[2]=2

Plot the approximate solution:

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-eow4xy
Out[3]=3

Compare with the exact solution:

In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-h0hg1j
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-g0ki7f
Out[5]=5

Find the approximate eigenfunctions for the Sturm–Liouville problem corresponding to , for large values of . Rewrite the problem using a small parameter :

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-boj8zw

Obtain a first-order asymptotic approximation:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-hmju7e
Out[2]=2

Construct the eigenfunctions after ignoring the constant Csc factor, using :

In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-b51uc2
Out[3]=3

Plot the approximate eigenfunctions:

In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-l82gy3
Out[4]=4

Properties & Relations  (3)Properties of the function, and connections to other functions

Solutions satisfy the differential equation up to a given order:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-c79dkx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bg2kw
Out[2]=2

Use DSolveValue to find an exact solution:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-idsn5b
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-vy6t8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-zjh66
Out[3]=3

Use NDSolveValue to find a numerical solution:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-ieoh19
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-gkyj6y
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-dr3kkp
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-fa72fc
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

The expansion returned for this example has fewer than four terms:

In[1]:=1
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-pr19ie
Out[1]=1

The missing terms are present in the expansion for a more general problem:

In[2]:=2
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-utoxn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bcnqmkihk6qf22lab6ka-bqfun6
Out[3]=3
Wolfram Research (2018), AsymptoticDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html (updated 2025).
✖
Wolfram Research (2018), AsymptoticDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html (updated 2025).

Text

Wolfram Research (2018), AsymptoticDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html (updated 2025).

✖
Wolfram Research (2018), AsymptoticDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html (updated 2025).

CMS

Wolfram Language. 2018. "AsymptoticDSolveValue." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html.

✖
Wolfram Language. 2018. "AsymptoticDSolveValue." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html.

APA

Wolfram Language. (2018). AsymptoticDSolveValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html

✖
Wolfram Language. (2018). AsymptoticDSolveValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html

BibTeX

@misc{reference.wolfram_2025_asymptoticdsolvevalue, author="Wolfram Research", title="{AsymptoticDSolveValue}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html}", note=[Accessed: 06-June-2025 ]}

✖
@misc{reference.wolfram_2025_asymptoticdsolvevalue, author="Wolfram Research", title="{AsymptoticDSolveValue}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html}", note=[Accessed: 06-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_asymptoticdsolvevalue, organization={Wolfram Research}, title={AsymptoticDSolveValue}, year={2025}, url={https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html}, note=[Accessed: 06-June-2025 ]}

✖
@online{reference.wolfram_2025_asymptoticdsolvevalue, organization={Wolfram Research}, title={AsymptoticDSolveValue}, year={2025}, url={https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html}, note=[Accessed: 06-June-2025 ]}