60.4.56 problem 1514

Internal problem ID [11479]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1514
Date solved : Sunday, March 30, 2025 at 08:23:03 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+\left (a \,x^{3}-12\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 129

ode:=x^3*diff(diff(diff(y(x),x),x),x)+6*x^2*diff(diff(y(x),x),x)+(a*x^3-12)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.541 (sec). Leaf size: 167

ode=(-12 + a*x^3)*y[x] + 6*x^2*D[y[x],{x,2}] + x^3*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \exp \left (-\int \frac {a^{2/3} x^2+4 \sqrt [3]{a} x+6}{\sqrt [3]{a} x^2+2 x} \, dx\right )+c_2 \exp \left (\int \frac {\sqrt [3]{-1} a^{2/3} x^2-4 \sqrt [3]{a} x-6 (-1)^{2/3}}{x \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )} \, dx\right )+c_3 \exp \left (\int \frac {-(-1)^{2/3} a^{2/3} x^2-4 \sqrt [3]{a} x+6 \sqrt [3]{-1}}{x \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )} \, dx\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 3)) + 6*x**2*Derivative(y(x), (x, 2)) + (a*x**3 - 12)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve x**3*Derivative(y(x), (x, 3)) + 6*x**2*Derivative(y(x), (x, 2)) + (a*x**3 - 12)*y(x)