54.4.27 problem 1483

Internal problem ID [12749]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1483
Date solved : Friday, October 03, 2025 at 03:47:18 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 37

ode:=2*x*diff(diff(diff(y(x),x),x),x)-4*(x+nu-1)*diff(diff(y(x),x),x)+(2*x+6*nu-5)*diff(y(x),x)+(1-2*nu)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{\frac {x}{2}} x^{\nu } \operatorname {BesselI}\left (\nu , \frac {x}{2}\right )+c_3 \,{\mathrm e}^{\frac {x}{2}} x^{\nu } \operatorname {BesselK}\left (\nu , \frac {x}{2}\right ) \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 74

ode=(1 - 2*nu)*y[x] + (-5 + 6*nu + 2*x)*D[y[x],x] - 4*(-1 + nu + x)*D[y[x],{x,2}] + 2*x*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (c_2 \int _1^xe^{-K[1]} \operatorname {HypergeometricU}\left (\nu -\frac {1}{2},2-2 \nu ,K[1]\right )dK[1]+c_3 \int _1^xe^{-K[2]} L_{\frac {1}{2}-\nu }^{1-2 \nu }(K[2])dK[2]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 3)) + (1 - 2*nu)*y(x) - (4*nu + 4*x - 4)*Derivative(y(x), (x, 2)) + (6*nu + 2*x - 5)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*nu*y(x) + 4*nu*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), (x, 2)) - 2*x*Derivative(y(x), (x, 3)) - y(x) - 4*Derivative(y(x), (x, 2)))/(6*nu + 2*x - 5) cannot be solved by the factorable group method