60.5.14 problem 1551

Internal problem ID [11511]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1551
Date solved : Sunday, March 30, 2025 at 08:23:57 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 62

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

\[ y = \frac {\left (c_4 \,\nu ^{2} x^{3}+6 c_4 \nu \,x^{2}+15 c_4 x +c_2 \right ) {\mathrm e}^{-\nu x}+{\mathrm e}^{\nu x} \left (c_3 \,\nu ^{2} x^{3}-6 c_3 \nu \,x^{2}+15 c_3 x +c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.431 (sec). Leaf size: 181

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

\[ y(x)\to \frac {c_4 x \exp \left (\int \frac {\nu ^3 x \left (x^3-1\right )+\nu ^2 \left (-4 x^3+6 x+1\right )+3 \nu \left (3 x^2-5 x-2\right )+15}{(x-1) x \left (\nu ^2 \left (x^2+x+1\right )-6 \nu (x+1)+15\right )} \, dx\right )+c_3 x \exp \left (\int \frac {\nu ^3 \left (x-x^4\right )+\nu ^2 \left (-4 x^3+6 x+1\right )+\nu \left (-9 x^2+15 x+6\right )+15}{(x-1) x \left (\nu ^2 \left (x^2+x+1\right )+6 \nu (x+1)+15\right )} \, dx\right )+c_1 e^{-\nu x-1}+c_2 e^{\nu x}}{x} \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve nu**2*(nu**2*x**2 + 4)*y(x) + x**2*Derivative(y(x), (x, 4)) - (2*nu**2*x**2 + 12)*Derivative(y(x), (x, 2))