60.4.32 problem 1488

Internal problem ID [11455]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1488
Date solved : Sunday, March 30, 2025 at 08:22:36 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 129

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

\[ y = \frac {-c_2 \left (\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i a^{3} x \right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}-c_3 \left (\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i a^{3} x \right ) {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2 a}}+c_1 \left (a^{3} x +2 \left (-a^{4}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} x}{a}}}{x} \]

✓ Mathematica. Time used: 0.473 (sec). Leaf size: 167

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

\[ y(x)\to c_1 \exp \left (-\int \frac {a^{2/3} x^2+2 \sqrt [3]{a} x+2}{\sqrt [3]{a} x^2+2 x} \, dx\right )+c_2 \exp \left (\int \frac {\sqrt [3]{-1} a^{2/3} x^2-2 \sqrt [3]{a} x-2 (-1)^{2/3}}{x \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )} \, dx\right )+c_3 \exp \left (\int \frac {-(-1)^{2/3} a^{2/3} x^2-2 \sqrt [3]{a} x+2 \sqrt [3]{-1}}{x \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )} \, dx\right ) \]

✗ Sympy

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

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