60.5.31 problem 1568

Internal problem ID [11528]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1568
Date solved : Wednesday, March 05, 2025 at 02:27:23 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+12 x^{2} y^{\prime \prime }+a y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 85

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+8*x^3*diff(diff(diff(y(x),x),x),x)+12*x^2*diff(diff(y(x),x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} x^{-\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{2} x^{\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_3 \,x^{-\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}+c_4 \,x^{\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 116

ode=a*y[x] + 12*x^2*D[y[x],{x,2}] + 8*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_1 x^{-\frac {1}{2} \sqrt {5-4 \sqrt {1-a}}}+c_2 x^{\frac {1}{2} \sqrt {5-4 \sqrt {1-a}}}+c_3 x^{-\frac {1}{2} \sqrt {4 \sqrt {1-a}+5}}+c_4 x^{\frac {1}{2} \sqrt {4 \sqrt {1-a}+5}}}{\sqrt {x}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x) + x**4*Derivative(y(x), (x, 4)) + 8*x**3*Derivative(y(x), (x, 3)) + 12*x**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : CRootOf is not supported over ZZ[a]