60.4.63 problem 1521

Internal problem ID [11486]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1521
Date solved : Sunday, March 30, 2025 at 08:23:24 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} \left (x +1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (3 x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

ode:=(1+x)*x^3*diff(diff(diff(y(x),x),x),x)-(4*x+2)*x^2*diff(diff(y(x),x),x)+(10*x+4)*x*diff(y(x),x)-4*(3*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\ln \left (x \right )^{2} c_3 x +c_2 \ln \left (x \right ) x +c_3 \,x^{2}+c_1 x +c_3 \right ) \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 158

ode=-4*(1 + 3*x)*y[x] + x*(4 + 10*x)*D[y[x],x] - x^2*(2 + 4*x)*D[y[x],{x,2}] + x^3*(1 + x)*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x^2 \left (c_2 \int _1^x\exp \left (\int _1^{K[4]}\frac {1}{K[1]^2+K[1]}dK[1]-\frac {1}{2} \int _1^{K[4]}\frac {2 K[2]+4}{K[2]^2+K[2]}dK[2]\right )dK[4]+c_3 \int _1^x\exp \left (\int _1^{K[5]}\frac {1}{K[1]^2+K[1]}dK[1]-\frac {1}{2} \int _1^{K[5]}\frac {2 K[2]+4}{K[2]^2+K[2]}dK[2]\right ) \int _1^{K[5]}\exp \left (-2 \int _1^{K[3]}\frac {1}{K[1]^2+K[1]}dK[1]\right )dK[3]dK[5]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*(x + 1)*Derivative(y(x), (x, 3)) - x**2*(4*x + 2)*Derivative(y(x), (x, 2)) + x*(10*x + 4)*Derivative(y(x), x) - (12*x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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