54.4.22 problem 1478

Internal problem ID [12744]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1478
Date solved : Friday, October 03, 2025 at 03:47:17 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime \prime }+3 y^{\prime \prime }-a \,x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 48

ode:=x*diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)-a*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{4}, \frac {5}{4}\right ], \frac {x^{4} a}{64}\right )+\frac {c_2 \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], \frac {x^{4} a}{64}\right )}{x}+c_3 x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {5}{4}, \frac {3}{2}\right ], \frac {x^{4} a}{64}\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 90

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

\begin{align*} y(x)&\to \frac {(2-2 i) c_1 \, _0F_2\left (;\frac {1}{2},\frac {3}{4};\frac {a x^4}{64}\right )}{\sqrt [4]{a} x}+c_2 \, _0F_2\left (;\frac {3}{4},\frac {5}{4};\frac {a x^4}{64}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [4]{a} c_3 x \, _0F_2\left (;\frac {5}{4},\frac {3}{2};\frac {a x^4}{64}\right ) \end{align*}

✗ Sympy

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

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