60.5.10 problem 1546

Internal problem ID [11507]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1546
Date solved : Thursday, March 13, 2025 at 08:53:30 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 126

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*a*x*diff(diff(diff(y(x),x),x),x)+6*a^2*x^2*diff(diff(y(x),x),x)+4*a^3*x^3*diff(y(x),x)+a^4*x^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x \left (2 \sqrt {3+\sqrt {6}}\, \sqrt {a}+2 \sqrt {3-\sqrt {6}}\, \sqrt {a}+a x \right )}{2}} \left (c_{2} {\mathrm e}^{\sqrt {a}\, x \left (\sqrt {3+\sqrt {6}}+2 \sqrt {3-\sqrt {6}}\right )}+c_4 \,{\mathrm e}^{\sqrt {a}\, x \left (2 \sqrt {3+\sqrt {6}}+\sqrt {3-\sqrt {6}}\right )}+c_3 \,{\mathrm e}^{\sqrt {3-\sqrt {6}}\, \sqrt {a}\, x}+c_{1} {\mathrm e}^{\sqrt {3+\sqrt {6}}\, \sqrt {a}\, x}\right ) \]

✓ Mathematica. Time used: 0.538 (sec). Leaf size: 165

ode=a^4*x^4*y[x] + 4*a^3*x^3*D[y[x],x] + 6*a^2*x^2*D[y[x],{x,2}] + 4*a*x*Derivative[3][y][x] + Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\frac {a x^2}{2}-\sqrt {3+\sqrt {6}} \sqrt {a} x} \left (6 a \left (c_1 e^{\frac {\left (-3+\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}+c_2 e^{\frac {\left (3+\sqrt {3}-\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}\right )+\sqrt {6} \sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )} \left (c_4 e^{\frac {2 a x}{\sqrt {a-\sqrt {\frac {2}{3}} a}}}+c_3\right )\right )}{6 a} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**4*x**4*y(x) + 4*a**3*x**3*Derivative(y(x), x) + 6*a**2*x**2*Derivative(y(x), (x, 2)) + 4*a*x*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*x*y(x)/4 + Derivative(y(x), x) + 3*Derivative(y(x), (x, 2))/(2*a*x) + Derivative(y(x), (x, 3))/(a**2*x**2) + Derivative(y(x), (x, 4))/(4*a**3*x**3) cannot be solved by the factorable group method