60.4.67 problem 1525

Internal problem ID [11490]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1525
Date solved : Sunday, March 30, 2025 at 08:23:28 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y&=0 \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 208

ode:=x^6*diff(diff(diff(y(x),x),x),x)+6*x^5*diff(diff(y(x),x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {4 \left (-c_3 \,{\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a x}} \left (\frac {\left (-a^{4}\right )^{{1}/{3}} \left (-i+\sqrt {3}\right )}{4}+i a x \right )-\left (\frac {\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{{1}/{3}}}{4}+i a x \right ) c_2 \,{\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a x}}+128 \,{\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}}}{a x}} c_1 \left (a x +\frac {\left (-a^{4}\right )^{{1}/{3}}}{2}\right )\right ) \left (-8 x^{3}+a \right )^{4}}{{\left (2 a x +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{{1}/{3}} \sqrt {3}-4 a x +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{{1}/{3}} \sqrt {3}+4 a x -\left (-a^{4}\right )^{{1}/{3}}\right )}^{4}} \]

✓ Mathematica. Time used: 0.517 (sec). Leaf size: 165

ode=a*y[x] + 6*x^5*D[y[x],{x,2}] + x^6*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x) + x**6*Derivative(y(x), (x, 3)) + 6*x**5*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a*y(x) + x**6*Derivative(y(x), (x, 3)) + 6*x**5*Derivative(y(x), (x, 2))