60.4.66 problem 1524

Internal problem ID [11489]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1524
Date solved : Sunday, March 30, 2025 at 08:23:27 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 95

ode:=x^6*diff(diff(diff(y(x),x),x),x)+x^2*diff(diff(y(x),x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2} \left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselK}\left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{{11}/{2}}}d x c_3 -\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (-2 x^{3} \operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{{11}/{2}}}d x c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 82

ode=-2*y[x] + x^2*D[y[x],{x,2}] + x^6*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x^2 \left (c_2 \int _1^x\frac {\left (-\frac {1}{3}\right )^{2/3} \operatorname {Hypergeometric1F1}\left (-\frac {2}{3},\frac {2}{3},\frac {1}{3 K[1]^3}\right )}{K[1]^2}dK[1]+c_3 \int _1^x-\frac {\operatorname {Hypergeometric1F1}\left (-\frac {1}{3},\frac {4}{3},\frac {1}{3 K[2]^3}\right )}{3 K[2]^3}dK[2]+c_1\right ) \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve x**6*Derivative(y(x), (x, 3)) + x**2*Derivative(y(x), (x, 2)) - 2*y(x)