60.4.45 problem 1501

Internal problem ID [11468]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1501
Date solved : Sunday, March 30, 2025 at 08:22:50 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.099 (sec). Leaf size: 114

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

\[ y(x)\to e^x \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {-2 \nu +2 K[1]+1}{2 K[1]}dK[1]\right ) \operatorname {HypergeometricU}\left (\nu -\frac {1}{2},2 \nu +1,K[2]\right )dK[2]+c_3 \int _1^x\exp \left (\int _1^{K[3]}-\frac {-2 \nu +2 K[1]+1}{2 K[1]}dK[1]\right ) L_{\frac {1}{2}-\nu }^{2 \nu }(K[3])dK[3]+c_1\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (4*nu**2*y(x) - 8*x**2*Derivative(y(x), (x, 2)) + 4*x**2*Derivative(y(x), (x, 3)) + 8*x*Derivative(y(x), (x, 2)) - y(x))/(4*nu**2 - 4*x**2 + 8*x - 1) cannot be solved by the factorable group method