Ordinary Differential Equation

Author

Eric W. Weisstein
October 10, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/OrdinaryDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Comments

To be broken off into individual notebooks

Abel

soln=DSolve[y''[x]+P[x] y'[x]+Q[x]y[x]==0,y[x],x]

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//Simplify

Airy

soln=DSolve[ode=y''[x]-x y[x]==0,y[x],x]

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//Simplify

Product of Airy functions

soln=FullSimplify[DSolve[ode=y'''[x]-4x y'[x]-2y[x]==0,y[x],x]]

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//Simplify

Anger

(soln=DSolve[ode=y''[x]+y'[x]/x+(1-ν^2/x^2)y[x]==(x-ν)/(π x^2)Sin[ν x],y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]

Baer

(soln=DSolve[ode=(x-(Subscript[a, 1]))(x-(Subscript[a, 2]))y''[x]+1/2(2x-((Subscript[a, 1])+(Subscript[a, 2])))y'[x]-(p^2x+q^2)y[x]==0,y[x],x])//Timing

Baer Wave Equation

(soln=DSolve[ode=(x-(Subscript[a, 1]))(x-(Subscript[a, 2]))y''[x]+1/2(2x-((Subscript[a, 1])+(Subscript[a, 2])))y'[x]-(k^2x^2-p^2x+q^2)y[x]==0,y[x],x])//Timing

Bernoulli

(soln=DSolve[ode=y'[x]+p[x]y[x]==q[x]y[x]^n,y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

Bessel

(soln=DSolve[ode=x^2 y''[x]+x y'[x]+(λ^2 x^2-n^2)y[x]==0,y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

Binomial

DSolve[y'[x]^m==f[x,y[x]],y[x],x]

Blasius

DSolve[y'''[x]+y[x]y''[x]==0,y[x],x]

Bôcher Equation

Briot-Bouquet

DSolve[x^m y'[x]==f[x,y[x]],y[x],x]

Brusselator

DSolve[{
u'[t]==a+u[t]^2v[t]-(b+1)u[t],
v'[t]==b u[t]-u[t]^2v[t]
},
{u[t],v[t]},t]//Timing

Full Brusselator

DSolve[{
u'[t]==1+u[t]^2v[t]-(w[t]+1)u[t],
v'[t]==u[t]w[t]-u[t]^2v[t],
w'[t]==-u[t]w[t]+α
},
{u[t],v[t],w[t]},t]//Timing

Chebyshev

4.1

DSolve[(1-x^2)y''[x]-x y'[x]+α^2 y[x]==0,y[x],x]

5.0

DSolve[(1-x^2)y''[x]-x y'[x]+α^2 y[x]==0,y[x],x]

5.2

(soln=DSolve[ode=(1-x^2)y''[x]-x y'[x]+α^2 y[x]==0,y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

Clairaut

(soln=DSolve[ode=y[x]==x y'[x]+f[y'[x]],y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

Complete Elliptic of the First Kind

v4.2

(soln=DSolve[ode=D[x(1-x^2)y'[x],x]-x y[x]==0,y[x],x])//Timing

v5.2

(soln=DSolve[ode=D[x(1-x^2)y'[x],x]-x y[x]==0,y[x],x])//Timing

FullSimplify[FunctionExpand[MeijerG[{{},{(1/2),(1/2)}},{{0,0},{}},(x^2)]]]

FullSimplify[D[x(1-x^2)y'[x],x]-x y[x]==0/.y->Function[x,((2 EllipticK[1-(x^2)])/π)]]

expr=ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

FunctionExpand[expr]

FullSimplify[FunctionExpand[(4 MeijerG[{{},{-(1/2),-(1/2)}},{{-1,0},{}},(x^2)]-4 MeijerG[{{},{-(1/2),-(1/2)}},{{-1,1},{}},(x^2)]+MeijerG[{{},{(1/2),(1/2)}},{{0,0},{}},(x^2)]-8 MeijerG[{{},{(1/2),(1/2)}},{{0,1},{}},(x^2)]+4 MeijerG[{{},{(1/2),(1/2)}},{{0,2},{}},(x^2)])]]

FullSimplify[FunctionExpand[expr]]

expr/.x->#&/@{-1,1}

Limit[Subtract@@expr,x->#]&/@{-1,1}

Table[FunctionExpand[expr]/.x->Random[Complex,5(I+1){-1,1}],{2}]//Chop//Union

Complete Elliptic of the Second Kind

v4.2

(soln=DSolve[ode=(1-x^2)D[x y'[x],x]+x y[x]==0,y[x],x])//Timing

v5.2

(soln=DSolve[ode=(1-x^2)D[x y'[x],x]+x y[x]==0,y[x],x])//Timing

FullSimplify[FunctionExpand[MeijerG[{{},{(1/2),(3/2)}},{{0,0},{}},(x^2)]]]

FullSimplify[Subtract@@{-(-1+(x^2)) Hypergeometric2F1[(1/2),(3/2),2,1-(x^2)],-((4 (EllipticE[1-(x^2)]-EllipticK[1-(x^2)]))/π)}]

FullSimplify[FunctionExpand[-(-1+(x^2)) Hypergeometric2F1[(1/2),(3/2),2,1-(x^2)]]]

FullSimplify[(1-x^2)D[x y'[x],x]+x y[x]==0/.y->Function[x,-((4 (EllipticE[1-(x^2)]-EllipticK[1-(x^2)]))/π)]]

expr=ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

FunctionExpand[expr]

FullSimplify[FunctionExpand[expr]]

FullSimplify[%30,Abs[x]≠1]

FullSimplify[%30,Abs[x]==1]

expr/.x->#&/@{-1,1}

Limit[Subtract@@expr,x->#]&/@{-1,1}

Table[FunctionExpand[expr]/.x->Random[Complex,5(I+1){-1,1}],{2}]//Chop//Union

Confluent Hypergeometric

(soln=DSolve[ode=x y''[x]+(γ-x)y'[x]-α y[x]==0,y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

Collect[Subtract@@%84,{C[1],C[2]}]

FullSimplify[(-α HypergeometricU[α,γ,x]+α (x-γ) HypergeometricU[1+α,1+γ,x]+x α (1+α) HypergeometricU[2+α,2+γ,x])]

Table[%96/.Thread[{α,γ,x}->Table[Random[Complex,5(I+1){-1,1},25],{3}]],{100}]//Chop//Union//Timing

FullSimplify[(x LaguerreL[-2-α,1+γ,x]+(x-γ) LaguerreL[-1-α,γ,x]-α LaguerreL[-α,-1+γ,x])]

Table[%/.Thread[{α,γ,x}->Table[Random[Complex,5(I+1){-1,1},25],{3}]],{100}]//Chop//Union//Timing

Coulomb

(soln=DSolve[ode=y''[x]+(1-2η/x-L(L+1)/x^2)y[x]==0,y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//Simplify

%//FullSimplify//Timing

d'Alembert's

DSolve[y[x]==x f[y'[x]]+g[y'[x]],y[x],x]

Duffing

v?

FullSimplify[DSolve[x''[t]+(Subscript[ω, 0])^2 x[t]+β x[t]^3==0,x[t],t]]//Timing

DSolve[y''[x]+y[x]+ϵ y[x]^3==0,y[x],x]

v5.2

(soln=DSolve[ode=x''[t]+(Subscript[ω, 0])^2 x[t]+β x[t]^3==0,x[t],t])//Timing

ode/.x->Function[t,Evaluate[soln[[1,1,2]]]]//FullSimplify

(soln=DSolve[ode=y''[x]+y[x]+ϵ y[x]^3==0,y[x],x])//Timing

ode/.y->Function[x,Evaluate[soln[[1,1,2]]]]//FullSimplify

Eckart

η=Exp[δ x];
DSolve[y''[x]+(((α η)/(1+η))+((β η)/((1+η)^2))+γ)y[x]==0,y[x],x]

Ellipsoidal Wave

DSolve[y''[x]-(a+b k^2 JacobiSN[x,k^2]^2+q k^4JacobiSN[x,k^2]^4)y[x]==0,y[x],x]

Emden-Fowler

DSolve[D[x^p y'[x],x]+x^σ y[x]^n==0,y[x],x]

Erfc ODE

DSolve[y''[x]+2x y'[x]-2n y[x]==0,y[x],x]

FullSimplify[y''[x]+2x y'[x]-2n y[x] /. y:>Function[x,(E^(-(x^2))) (C[1] HermiteH[-1-n,x])]]

FullSimplify[y''[x]+2x y'[x]-2n y[x] /. y:>Function[x,(E^(-(x^2))) (C[2] Hypergeometric1F1[((1+n)/2),(1/2),(x^2)])]]

erfc[n_,z_]:=(2^(-n)) (E^(-(z^2))) (-((2 z Hypergeometric1F1[1+(n/2),(3/2),(z^2)])/(Gamma[((1+n)/2)]))+((Hypergeometric1F1[((1+n)/2),(1/2),(z^2)])/(Gamma[1+(n/2)])))

FullSimplify[y''[x]+2x y'[x]-2n y[x]/. y->Function[x,C[1]  erfc[n,x]],n∈Integers]//Timing

FullSimplify[y''[x]+2x y'[x]-2n y[x]/. y->Function[x,C[2]  erfc[n,-x]],n∈Integers]//Timing

f[n_,x_]:=(2 (E^(-(x^2))) (2 (2+n) x (C[1]-C[2]) Gamma[1+(n/2)] (15 Hypergeometric1F1[1+(n/2),(3/2),(x^2)]+5 (-3+2 (x^2)) Hypergeometric1F1[2+(n/2),(5/2),(x^2)]-2 (4+n) (x^2) Hypergeometric1F1[3+(n/2),(7/2),(x^2)])+5 (1+n) (C[1]+C[2]) Gamma[((1+n)/2)] (-3 Hypergeometric1F1[((1+n)/2),(1/2),(x^2)]+(3-6 (x^2)) Hypergeometric1F1[((3+n)/2),(3/2),(x^2)]+2 (3+n) (x^2) Hypergeometric1F1[((5+n)/2),(5/2),(x^2)])))/(15 (Sqrt[π]) Gamma[1+n])

Plot[f[2,x]/.{C[1]->1,C[2]->1},{x,-1,1}]

Euler

Usual Type

DSolve[x^2 y''[x]+a x y'[x]+b y[x]==0,y[x],x]

DSolve[x^2 y''[x]+a x y'[x]+b y[x]==S[x],y[x],x]

Valiron, p. 201

DSolve[y'[x]+Sqrt[(a y[x]^4+b y[x]^3+c y[x]^2+d y[x]+e)/(a x^4+b x^3+c x^2+d x+e)]==0,y[x],x]

Valiron, p. 212

FullSimplify[DSolve[y'[x]+y[x]^2-α x^m==0,y[x],x]]

Exponential

DSolve[x'[t]==k x[t],x[t],t]

DSolve[x'[t]^2==k^2 x[t]^2,x[t],t]

DSolve[x'[t]==k Abs[x[t]],x[t],t]//Timing

DSolve[x'[t]==k Abs[x[t]],x[t],t]//Timing

Falkner-Skan

DSolve[y'''[x]+α y[x]y''[x]+β(1-y'[x]^2)==0,y[x],x]//Timing

Fractional

DSolve[D[y[t],{t,2 v}]+a D[y[t],{t,v}]+b y[t]==0,y[t],t]

General Confluent Hypergeometric Equation

Abramowitz and Stegun, p. 505

(((f'(x))^2)+((2 A f'(x))/x)+((A (A-1))/(x^2))-((a ((h'(x))^2))/(h(x)))+((A/x)+f'(x)) (((h'(x) b)/(h(x)))-h'(x)-((h''(x))/(h'(x))))+f''(x)) y(x)+(((2 A)/x)+((b h'(x))/(h(x)))+2 f'(x)-h'(x)-((h''(x))/(h'(x)))) y'(x)+y''(x)

4.1

DSolve[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x]==0,y[x],x]

FullSimplify[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x] /. y:>Function[x,x^(-A)Exp[-f[x]]Hypergeometric1F1[a,b,h[x]]]]

FullSimplify[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x] /. y:>Function[x,x^(-A)Exp[-f[x]]HypergeometricU[a,b,h[x]]]]

4.2

DSolve[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x]==0,y[x],x]

FullSimplify[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x] /. y:>Function[x,x^(-A)Exp[-f[x]]Hypergeometric1F1[a,b,h[x]]]]

FullSimplify[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x] /. y:>Function[x,x^(-A)Exp[-f[x]]HypergeometricU[a,b,h[x]]]]

5.0

DSolve[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x]==0,y[x],x]

FullSimplify[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x] /. y:>Function[x,x^(-A)Exp[-f[x]]Hypergeometric1F1[a,b,h[x]]]]

FullSimplify[y''[x]+(2A/x+2f'[x]+b h'[x]/h[x]-h'[x]-h''[x]/h'[x])y'[x]+((b h'[x]/h[x]-h'[x]-h''[x]/h'[x])(A/x+f'[x])+A(A-1)/x^2+2A f'[x]/x+f''[x]+f'[x]^2-a h'[x]^2/h[x])y[x] /. y:>Function[x,x^(-A)Exp[-f[x]]HypergeometricU[a,b,h[x]]]]

Halm's

DSolve[(1+x^2)^2 y''[x]+λ y[x]==0,y[x],x]

Heine

DSolve[y''[x]+(1/2)((1/(x-(Subscript[a, 1])))+(2/(x-(Subscript[a, 3]))))y'[x]+(1/4)((((Subscript[A, 0])+(Subscript[A, 1])x+(Subscript[A, 2])x^2+(Subscript[A, 3])x^3)/((x-(Subscript[a, 1]))(x-(Subscript[a, 2]))^2(x-(Subscript[a, 3]))^2)))y[x]==0,y[x],x]

Hermite

DSolve[y''[x]-2x y'[x]==0,y[x],x]

c1 Integrate[1/ E^(-x^2),x]+c2

DSolve[y''[x]-2x y'[x]+λ y[x]==0,y[x],x]

Odd Terms

Product[(4k-λ),{k,0,n}]

FunctionExpand[%]

Simplify[Series[Sum[(4^n) Pochhammer[-(λ/4),n]/(2n)!x^(2n),{n,3}],{x,0,4}]]

FullSimplify[1+Sum[(4^n) Pochhammer[-(λ/4),n]/(2n)!x^(2n),{n,Infinity}]]

FunctionExpand[Hypergeometric1F1[-(λ/4),(1/2),(x^2)]]

Even Terms

Product[(2+4k-λ),{k,0,n}]

FunctionExpand[%]

Simplify[Series[Sum[(4^n) Pochhammer[-(1/4) (-2+λ),n]/(2n+1)!x^(2n+1),{n,3}],{x,0,5}]]

FullSimplify[x+Sum[(4^n) Pochhammer[-(1/4) (-2+λ),n]/(2n+1)!x^(2n+1),{n,Infinity}]]

FunctionExpand[x Hypergeometric1F1[-(1/4) (-2+λ),(3/2),(x^2)]]

Full Solution

soln=(Subscript[a, 0])Hypergeometric1F1[-(λ/4),(1/2),(x^2)]+(Subscript[a, 1])x Hypergeometric1F1[-(1/4) (-2+λ),(3/2),(x^2)];

Simplify[Series[soln,{x,0,6}]]

FullSimplify[Table[soln /. λ->n,{n,0,4,2}]]//ColumnForm

Product[-4k+4n,{n,0,j-1}]

Simplify[Table[(4^j) Pochhammer[-k,j],{j,5}] /. k->λ/4]

Simplify[Table[(4^j) Pochhammer[-k,j],{j,5}] /. k->(λ-2)/4]

FullSimplify[(-1)^k 2^k(2k-1)!!(1+Sum[(4^j) Pochhammer[-k,j]x^(2j)/(2j)!,{j,k}])]

Simplify[Table[(((-2))^k) (-1+2 k)!! Hypergeometric1F1[-k,(1/2),(x^2)]-HermiteH[2k,x],{k,5}]]

FullSimplify[(-1)^k 2^(k+1)(2k+1)!!(x+Sum[(4^j) Pochhammer[-k,j]x^(2j+1)/(2j+1)!,{j,k}])]

Simplify[Table[(((-1))^k) (2^(1+k)) x (1+2 k)!! Hypergeometric1F1[-k,(3/2),(x^2)]-HermiteH[2k+1,x],{k,5}]]

Heun

DSolve[w''[x]+((γ/x)+(δ/(x-1))+(ϵ/(x-a)))w'[x]+((α β x-q)/(x(x-1)(x-a)))w[x]==0,w[x],x]

Hill

DSolve[y''[x]+(Subscript[θ, 0]) y[x]==0,y[x],x]

DSolve[y''[x]+((Subscript[θ, 0])+(Subscript[θ, 1]) Cos[2x])y[x]==0,y[x],x]

DSolve[y''[x]+((Subscript[θ, 0])+(Subscript[θ, 1]) Cos[2x]+(Subscript[θ, 2])Cos[4x])y[x]==0,y[x],x]

TrigExpand[Cos[4x]]

Hypergeometric

Mac

$Version

DSolve[x(x-1)y''[x]+((1+α+β)x-γ)y'[x]+α β y[x]==0,y[x],x]//Timing

Linux

$Version

DSolve[x(x-1)y''[x]+((1+α+β)x-γ)y'[x]+α β y[x]==0,y[x],x]//Timing

Jacobi

Physics Version

Mac

$Version

DSolve[(1-x^2)y''[x]+(β-α-(α+β+2)x)y'[x]+n(n+α+β+1)y[x]==0,y[x],x]//Timing

f=FullSimplify[(1-x^2)y''[x]+(β-α-(α+β+2)x)y'[x]+n(n+α+β+1)y[x] /.y->Function[x,JacobiP[n,α,β,x]]]

Plot[f /.{α->1.1,β->2.1,n->2},{x,-1,1}]

Linux

$Version

DSolve[(1-x^2)y''[x]+(β-α-(α+β+2)x)y'[x]+n(n+α+β+1)y[x]==0,y[x],x]//Timing

Zwillinger p. 120 (Ince)

DSolve[((Subscript[a, 1])+(Subscript[b, 1])x+(Subscript[c, 1])y[x])(x y'[x]-y[x])-((Subscript[a, 2])+(Subscript[b, 2])x+(Subscript[c, 2])y[x])y'[x]+((Subscript[a, 3])+(Subscript[b, 3])x+(Subscript[c, 3])y[x])==0,y[x],x]//Timing

Zwillinger p. 123 (Iyanaga and Kawada)

FullSimplify[DSolve[x(1-x)y''[x]+(γ-(α+1)x)y'[x]+n(α+n)y[x]==0,y[x],x]]

Jacobi Elliptic

DSolve[{
u'[t]==v[t]w[t],
v'[t]==-u[t]w[t],
w'[t]==-k^2 u[t]v[t]},
{u[t],v[t],w[t]},t]

{u'[t]==v[t]w[t],
v'[t]==-u[t]w[t],
w'[t]==-k^2 u[t]v[t]}/.{
u->Function[t,JacobiSN[t,k^2]],
v->Function[t,JacobiCN[t,k^2]],
w->Function[t,JacobiDN[t,k^2]]}

Kelvin

FullSimplify[DSolve[x^2 y''[x]+x y'[x]-(I x^2+ν^2)y[x]==0,y[x],x]]

FullSimplify[x^2 y''[x]+x y'[x]-(I x^2+ν^2)y[x]/.y->Function[x,BesselJ[ν,-(((-1))^(3/4)) x] C[1]+BesselY[ν,-(((-1))^(3/4)) x] C[2]]]//Timing

ToRadicals[RootReduce[ExpToTrig[Exp[-Pi I/4]]]]

Re[BesselJ[n,x Exp[3Pi I/4]]]

Lagerstrom

DSolve[y''[x]+k/x y'[x]+ϵ y[x]y'[x]==0,y[x],x]//Timing

Laguerre

DSolve[x y''[x]+(1-x)y'[x]+λ y[x]==0,y[x],x]

DSolve[x y''[x]+(ν+1-x)y'[x]+λ y[x]==0,y[x],x]

λ = 0

DSolve[x y''[x]+(ν+1-x)y'[x]+0 y[x]==0,y[x],x]

FullSimplify[C[1]Integrate[Exp[x]/x^(ν+1),x]+C[2]]

Lamé

Byerly

DSolve[(x^2-b^2)(x^2-c^2)z''[x]+x(x^2-b^2+x^2-c^2)z'[x]
-(m(m+1)x^2-(b^2+c^2)p)z[x]==0,z[x],x]

Zwillinger, p. 124 (Ward)

DSolve[y''[x]+(h-n(n+1)k^2JacobiSN[x,k^2])y[x]==0,y[x],x]

Zwillinger, p. 124 (Moon and Spencer)

DSolve[y''[x]+(1/2)((1/(x-(Subscript[a, 1])))+(1/(x-(Subscript[a, 2])))+(1/(x-(Subscript[a, 3]))))y'[x]+(1/4)((((Subscript[A, 0])+(Subscript[A, 1])x)/((x-(Subscript[a, 1]))(x-(Subscript[a, 2]))(x-(Subscript[a, 3])))))y[x]==0,y[x],x]

Zwillinger, p. 124 (Moon and Spencer), wave equation

DSolve[y''[x]+(1/2)((1/x)+(1/(x-(Subscript[a, 2])))+(1/(x-(Subscript[a, 3]))))y'[x]+(1/4)((((((Subscript[a, 2])^2)+((Subscript[a, 3])^2))q-p(p+1)x+κ x^2)/(x(x-(Subscript[a, 2]))(x-(Subscript[a, 3])))))y[x]==0,y[x],x]

Lane-Emden

Legendre

Nonassociated

DSolve[(1-x^2)y''[x]-2x y'[x]+l(l+1)y[x]==0,y[x],x]//Timing

Associated

DSolve[(1-x^2)y''[x]-2x y'[x]+(l(l+1)-m^2/(1-x^2))y[x]==0,y[x],x]//Timing

Wave Equation

DSolve[(1-x^2)y''[x]-2x y'[x]+(k^2 a^2(x^2-1)-p(p+1)-((q^2)/(x^2-1)))y[x]==0,y[x],x]//Timing

Lewis Regulator

DSolve[y''[x]+(1-Abs[y[x]])y'[x]+y[x]==0,y[x],x]//Timing

DSolve[y''[x]+(1-y[x])y'[x]+y[x]==0,y[x],x]//Timing

DSolve[y''[x]+(1+y[x])y'[x]+y[x]==0,y[x],x]//Timing

Liénard's Equaton

DSolve[y''[x]+f[x]y'[x]+y[x]==0,y[x],x]

Liouville's

DSolve[y''[x]+g[y[x]]y'[x]^2+f[x]y'[x]==0,y[x],x]

Lommel

DSolve[z^2 y''[z]+z y'[z]-(z^2+ν^2)y[z]==k z^(μ+1),y[z],z]

Lorenz

DSolve[{
X'[t]==σ(Y[t]-X[t]),
Y'[t]==-X[t]Z[t]+r X[t]-Y[t],
Z'[t]==X[t]Y[t]-b Z[t]},
{X[t],Y[t],Z[t]},t]

Lotka-Volterra

Löwner's

DSolve[y'[x]==-y[x](1+κ[x]y[x])/(1-κ[x]y[x]),y[x],x]//Timing

Lotka-Volterra

DSolve[{x'[t]==a x[t]-b x[t]y[t],y'[t]==-c y[t]+d x[t]y[t]},{x[t],y[t]},t]

Magnetic Pole Equation

DSolve[y''[x]-(((m(m+1)+1/4-(m+1/2)Cos[x])/(Sin[x]^2))+(λ+1/2))y[x]==0,y[x],x]

FullSimplify[-((1/2)+λ+((1/4)+m (1+m)-((1/2)+m) Cos[x]) ((Csc[x])^2)) y[x]+y''[x]]

DSolve[(1/4) (-2-4 λ+(1+2 m) (-1-2 m+2 Cos[x]) ((Csc[x])^2)) y[x]+y''[x]==0,y[x],x]

Mathieu

Normal

DSolve[V''[v]+(a-2q Cos[2v])V[v]==0,V[v],v]

Associated

DSolve[y''[x]+((1-2r)Cot[x])y'[x]+(a+k^2 Cos[x]^2)y[x]==0,y[x],x]

Modified

DSolve[y''[x]-(a-2q Cosh[2x])y[x]==0,y[x],x]

Modified Bessel

DSolve[x^2 y''[x]+x y'[x]-(x^2+n^2)y[x]==0,y[x],x]

FullSimplify[x^2 y''[x]+x y'[x]-(x^2+n^2)y[x]/. {y[x_]:>BesselI[n,x] (Subscript[C, 1])+BesselK[n,x] (Subscript[C, 2]),
Derivative[k_][y_][x_]:>D[BesselI[n,x] (Subscript[C, 1])+BesselK[n,x] (Subscript[C, 2]),{x,k}]}]

FullSimplify[x^2 y''[x]+x y'[x]-(x^2+n^2)y[x]/. {
y[x_]:>BesselJ[n,-I x] (Subscript[C, 1])+BesselY[n,-I x] (Subscript[C, 2]),
Derivative[k_][y_][x_]:>D[BesselJ[n,-I x] (Subscript[C, 1])+BesselY[n,-I x] (Subscript[C, 2]),{x,k}]
}]

n = 0

DSolve[x^2 y''[x]+x y'[x]-x^2 y[x]==0,y[x],x]

DSolve[D[x y'[x],x]==x y[x],y[x],x]

Modified Spherical Bessel

DSolve[r^2 R''[r]+2r R'[r]-(k^2 r^2+n(n+1))R[r]==0,R[r],r]

FunctionExpand[%]

Morse-Rosen

DSolve[y''[x]+((α/(Cosh[a x]^2))+β Tanh[a x]+γ)y[x]==0,y[x],x]

Neumann

DSolve[x^2 y''[x]+3x y'[x]+(x^2+1-n^2)y[x]==x Cos[n Pi/2]^2+n Sin[n Pi/2]^2,y[x],x]

FullSimplify[%,n∈Integers]

Simplify[%,n∈Integers]

FullSimplify[Table[DSolve[x^2 y''[x]+3x y'[x]+(x^2+1-n^2)y[x]==x Cos[n Pi/2]^2+n Sin[n Pi/2]^2,y[x],x],{n,0,1}]]

NeumannOSum[0,z_]:=1/z
NeumannOSum[n_,z_]:=Sum[n(n-m-1)!/m!(z/2)^(2m-n-1),{m,0,Floor[n/2]}]/4

t=Table[NeumannOSum[n,x],{n,0,5}]

Simplify[#[[1]]/.#[[2]]&/@Table[{x^2 y''[x]+3x y'[x]+(x^2+1-n^2)y[x]-(x Cos[n Pi/2]^2+n Sin[n Pi/2]^2) ,y->Function[x,Evaluate[t[[n+1]]]]},{n,0,4}]]

Oblate Spheroidal Wave Function

Angular

DSolve[D[(1-x^2)y'[x],x]+(λ+c^2x^2-((m^2)/(1-x^2)))y[x]==0,y[x],x]

Radial

DSolve[D[(1+x^2)y'[x],x]-(λ-c^2x^2-((m^2)/(1+x^2)))y[x]==0,y[x],x]

Onsager Equation

DSolve[D[Exp[x]D[Exp[x]y'[x],{x,2}],{x,3}]==f[x],y[x],x]

Orr-Sommerfeld Equation

(1/(I α R))(((((d^2)/(dx^2))-(α^2)))^2)y-((f[x]-c)(((d^2)/(dx^2))-(α^2))-f''[x])y[x]==0

Parabolic Cylinder Equation

General

FullSimplify[DSolve[y''[x]+(a x^2+b x+c)y[x]==0,y[x],x]]

FullSimplify[DSolve[y''[x]+(a x^2+b x+c)y[x]==0,y[x],x]]

Specific

FullSimplify[DSolve[y''[x]+(n+1/2-x^2/4)y[x]==0,y[x],x]]

Poisson-Boltzmann

DSolve[y''[x]+k/x y'[x]==-δ Exp[y[x]],y[x],x]

Pöschl-Teller

First

DSolve[y''[x]-(a^2(((κ(κ-1))/(Sin[a x]^2))+((λ(λ-1))/(Cos[a x]^2)))-b^2)y[x]==0,y[x],x]

Second

DSolve[y''[x]-(a^2(((κ(κ-1))/(Sinh[a x]^2))+((λ(λ-1))/(Cosh[a x]^2)))-b^2)y[x]==0,y[x],x]

Rayleigh

DSolve[y''[x]-μ(1-1/3y'[x]^2)y'[x]+y[x]==0,y[x],x]//Timing

DSolve[y''[x]-μ(1-y'[x]^2)y'[x]+y[x]==0,y[x],x]//Timing

Riccati

DSolve[w'[x]==q0[x]+q1[x]w[x]+q2[x]w[x]^2,w[x],x]//Timing

DSolve[x^2 y''[x]+(x^2-n(n+1))y[x]==0,y[x],x]

Riemann P-Differential

Mac

$Version

DSolve[u''[z]+(((1-α-α')/(z-a))+((1-β-β')/(z-b))+((1-γ-γ')/(z-c)))u'[z]+(((α α'(a-b)(a-c))/(z-a))+((β β'(b-c)(b-a))/(z-b))+((γ γ'(c-a)(c-b))/(z-c)))((u[z])/((z-a)(z-b)(z-c)))==0,u[z],z]//Timing

Aborted after running overnight

Liunx

$Version

DSolve[u''[z]+(((1-α-α')/(z-a))+((1-β-β')/(z-b))+((1-γ-γ')/(z-c)))u'[z]+(((α α'(a-b)(a-c))/(z-a))+((β β'(b-c)(b-a))/(z-b))+((γ γ'(c-a)(c-b))/(z-c)))((u[z])/((z-a)(z-b)(z-c)))==0,u[z],z]//Timing

Sharpe's differential equation

DSolve[z y''[z]+y'[z]+(z+A)y[z]==0,y[z],z]

Simple Harmonic Motion

DSolve[y''[x]+(Subscript[ω, 0])^2 y[x]==0,y[x],x]

Simple Harmonic Motion—Damped

DSolve[y''[x]+β y'[x]+(Subscript[ω, 0])^2 y[x]==0,y[x],x]

Simple Harmonic Motion—Forced and Damped

DSolve[y''[x]+β y'[x]+(Subscript[ω, 0])^2 y[x]==A Cos[ω t],y[x],x]

Simple Harmonic Motion—Quadratic Perturbation

DSolve[x''[t]+(Subscript[ω, 0])^2 x[t]-α ϵ x[t]^2==0,x[t],t]//Timing

Spherical Bessel

DSolve[r^2 R''[r]+2r R'[r]+(k^2 r^2-n(n+1))R[r]==0,R[r],r]

Struve

DSolve[z^2 y''[z]+z y'[z]+(z^2-ν^2)y[z]==((4(z/2)^(ν+1))/(Sqrt[Pi]Gamma[ν+1/2])),y[z],z]

FullSimplify[%]//Timing

FullSimplify[z^2 y''[z]+z y'[z]+(z^2-ν^2)y[z] /.y->Function[z,(1/2) z Csc[π ν] (-(2 (z^(2 ν)) BesselJ[-ν,z] HypergeometricPFQ[{(1/2)+ν},{1+ν,(3/2)+ν},-((z^2)/4)])/Gamma[2+2 ν]+(π BesselJ[ν,z] HypergeometricPFQRegularized[{(1/2)},{(3/2),1-ν},-((z^2)/4)])/Gamma[(1/2)+ν])]]//Timing

Sturm-Liouville

DSolve[D[p[x]y'[x],x]+(λ w[x]-q[x])y[x]==0,y[x],x]

Symmetrc Top

DSolve[y''[x]-(((m^2-1/4+k^2-2m k Cos[x])/(Sin[x]^2))+(σ+k^2+1/4))y[x]==0,y[x],x]

Thomas-Fermi

DSolve[y''[x]==y[x]^(3/2)x^(-1/2),y[x],x]

Titchmarsh's Equation

DSolve[y''[x]+(λ-x^(2n))y[x]==0,y[x],x]

Ultraspherical

Mathematica's Gegenbauer definition

This form of the solution is for μ not an integer

soln=DSolve[(1-x^2)y''[x]-(2μ+1)x y'[x]+ν(ν+2μ)y[x]==0,y[x],x][[1,1,2]]

soln=FullSimplify[(1-x^2)y''[x]-(2μ+1)x y'[x]+ν(ν+2μ)y[x] /. y:>Function[x,C[1] GegenbauerC[ν,μ,x]]]/C[1]

FunctionExpand[soln]

Plot[soln/.{ν->2.2,μ->3.3},{x,-1,1}]

μ an explici integer

soln=DSolve[Evaluate[(1-x^2)y''[x]-(2μ+1)x y'[x]+ν(ν+2μ)y[x]==0 /.μ->1],y[x],x][[1,1,2]]

FullSimplify[(1-x^2)y''[x]-(2μ+1)x y'[x]+ν(ν+2μ)y[x] /. y:>Function[x,((C[2] LegendreQ[(1/2) (1+2 ν),(1/2),x])/(((-1+(x^2)))^(1/4)))]/.μ->1]

Plot[(LegendreQ[-(3/2)+ν,(1/2),x]-2 x LegendreQ[-(1/2)+ν,(1/2),x]+LegendreQ[(1/2)+ν,(1/2),x])/.ν->.3,{x,-1,1}]

LegendreP Solution

soln1=FullSimplify[(1-x^2)y''[x]-(2μ+1)x y'[x]+ν(ν+2μ)y[x] /. y:>Function[x,(((-1+(x^2)))^((1/4) (1-2 μ))) C[1] LegendreP[(1/2) (-1+2 μ+2 ν),(1/2) (1-2 μ),x]]]

Simplify@Table[soln1 ,{μ,2},{ν,2}]

FullSimplify@Table[soln1 ,{μ,1/2,2},{ν,1/2,2}]

Plot[ (-4 EllipticE[((1-x)/2)]+2 EllipticK[((1-x)/2)]+π LegendreP[(1/2),x]),{x,-1,1}]

LegendreQ Solution

soln2=FullSimplify[(1-x^2)y''[x]-(2μ+1)x y'[x]+ν(ν+2μ)y[x] /. y:>Function[x,(((-1+(x^2)))^((1/4) (1-2 μ))) C[2] LegendreQ[(1/2) (-1+2 μ+2 ν),(1/2) (1-2 μ),x]]]

Simplify@Table[soln2 ,{μ,2},{ν,2}]

Plot[(LegendreQ[(1/2),-(1/2),x]-4 x LegendreQ[(3/2),-(1/2),x]+3 LegendreQ[(5/2),-(1/2),x]),{x,-1,1}]

Simplify@Table[soln2 ,{μ,1,2},{ν,1/2,2}]

Plot[(LegendreQ[-1,-(1/2),x]+2 x LegendreQ[0,-(1/2),x]-3 LegendreQ[1,-(1/2),x]),{x,-1,1}]

Simplify@Table[soln2 ,{μ,1/2,5},{ν,1,3}]

Simplify@Table[soln2 ,{μ,1/2,2},{ν,1/2,2}]

Plot[(LegendreQ[-(3/2),x]-LegendreQ[(1/2),x]),{x,-1,1}]

Hyperspherical (Zwillinger and Iyanaga and Kawada)

FullSimplify[DSolve[(1-x^2)y''[x]-2(μ+1)x y'[x]+(ν-μ)(ν+μ+1)y[x]==0,y[x],x]]

TraditionalForm[%]

Gegenbauer (Zwillinger and Infeld/Hull)

FullSimplify[DSolve[(1-x^2)y''[x]-(2m+3)x y'[x]+λ y[x]==0,y[x],x]]

van der Pol

DSolve[y''[x]-μ(1-y[x]^2)y'[x]+y[x]==0,y[x],x]//Timing

Wangerin

DSolve[y''[x]+(1/2)((1/(x-(Subscript[a, 1])))+(1/(x-(Subscript[a, 2])))+(1/(x-(Subscript[a, 3]))))y'[x]+(1/4)((((Subscript[A, 0])+(Subscript[A, 1])x+(Subscript[A, 2])x^2)/((x-(Subscript[a, 1]))(x-(Subscript[a, 2]))(x-(Subscript[a, 3])))))y[x]==0,y[x],x]//Timing

Weber

Spencer and Moon (Zwillinger 1997, p. 128)

FullSimplify[DSolve[y''[x]+(λ^2-x^2/4)y[x]==0,y[x],x]]

Others

FullSimplify[DSolve[w''[z]+(n+1/2-z^2/4)w[z]==0,w[z],z]]

DSolve[U''[u]-(c+k^2 u^2)U[u]==0,U[u],u]

DSolve[V''[v]+(c-k^2 v^2)V[v]==0,V[v],v]

Gradshteyn and Ryzhik, Zwillinger 1997, p. 127

DSolve[y''[x]+y'[x]/x+(1-((ν^2)/(x^2)))y[x]==-(1/(π (x^2)))(x+ν+(x-ν)Cos[ν  π]),y[x],x]

Weierstrass

DSolve[y'[x]==Sqrt[4y[x]^3-(Subscript[g, 2])y[x]-(Subscript[g, 3])],y[x],x]

Whittaker

Untransformed

FullSimplify[DSolve[u''[z]+u'[z]+((k/z)+((1/4-m^2)/(z^2)))u[z]==0,u[z],z]]

Transformed

FullSimplify[DSolve[y''[x]+(-1/4+(k/x)+((1/4-m^2)/(x^2)))y[x]==0,y[x],x]]