60.4.19 problem 1472

Internal problem ID [11442]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1472
Date solved : Sunday, March 30, 2025 at 08:22:19 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 33

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

\[ y = \left (\int \left (c_1 +c_2 \int {\mathrm e}^{-\int \frac {x^{3} f \left (x \right )+3}{x}d x}d x \right )d x +c_3 \right ) x \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 85

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

\[ y(x)\to x \left (c_3 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}f(K[1]) K[1]^2dK[1]\right )}{K[2]^2}dK[2]-x \int _1^x\frac {\exp \left (-\int _1^{K[3]}f(K[1]) K[1]^2dK[1]\right )}{K[3]^3}dK[3]\right )+c_2 x+c_1\right ) \]

✗ Sympy

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

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