60.5.13 problem 1550

Internal problem ID [11510]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1550
Date solved : Sunday, March 30, 2025 at 08:23:56 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.126 (sec). Leaf size: 139

ode:=x*diff(diff(diff(diff(y(x),x),x),x),x)-(6*x^2+1)*diff(diff(diff(y(x),x),x),x)+12*x^3*diff(diff(y(x),x),x)-(9*x^2-7)*x^2*diff(y(x),x)+2*(x^2-3)*x^3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x^{2}} \left (-\int \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right )}{x^{{3}/{2}}}d x c_4 -\int \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right )}{x^{{3}/{2}}}d x c_3 +c_1 \right )+{\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{4}} \operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right )}{x^{{3}/{2}}}d x c_4 +\int \frac {{\mathrm e}^{\frac {x^{2}}{4}} \operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right )}{x^{{3}/{2}}}d x c_3 +c_2 \right ) \]

✓ Mathematica. Time used: 1.211 (sec). Leaf size: 220

ode=2*x^3*(-3 + x^2)*y[x] - x^2*(-7 + 9*x^2)*D[y[x],x] + 12*x^3*D[y[x],{x,2}] - (1 + 6*x^2)*Derivative[3][y][x] + x*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\frac {x^2}{2}} \left (c_3 \int _1^x\frac {e^{\frac {K[1]^2}{2}} \left (\int \frac {e^{\frac {1}{4} \left (-1+\sqrt {5}\right ) K[1]^2-1} \operatorname {HypergeometricU}\left (-\frac {1}{4}+\frac {9}{4 \sqrt {5}},-\frac {1}{2},-\frac {1}{2} \sqrt {5} K[1]^2\right ) \left (K[1]^2\right )^{3/4}}{K[1]^{7/2}} \, dK[1]\right ) K[1]}{\sqrt [4]{2}}dK[1]+c_4 \int _1^x\frac {e^{\frac {K[2]^2}{2}} \left (\int \frac {e^{\frac {1}{4} \left (-1+\sqrt {5}\right ) K[2]^2-1} \left (K[2]^2\right )^{3/4} L_{\frac {1}{4}-\frac {9}{4 \sqrt {5}}}^{-\frac {3}{2}}\left (-\frac {1}{2} \sqrt {5} K[2]^2\right )}{K[2]^{7/2}} \, dK[2]\right ) K[2]}{\sqrt [4]{2}}dK[2]+c_2 e^{\frac {x^2}{2}}+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*(2*x**2 - 6)*y(x) + 12*x**3*Derivative(y(x), (x, 2)) - x**2*(9*x**2 - 7)*Derivative(y(x), x) + x*Derivative(y(x), (x, 4)) - (6*x**2 + 1)*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x**5*y(x) - 6*x**3*y(x) + 12*x**3*Derivative(y(x), (x, 2)) - 6*x**2*Derivative(y(x), (x, 3)) + x*Derivative(y(x), (x, 4)) - Derivative(y(x), (x, 3)))/(x**2*(9*x**2 - 7)) cannot be solved by the factorable group method