60.4.65 problem 1523

Internal problem ID [11488]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1523
Date solved : Sunday, March 30, 2025 at 08:23:26 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 150

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

\[ y(x)\to \frac {1}{2} x \left (-2 c_3 \left (x^2-3 x+1\right ) \int _1^x\frac {\left (\int \exp \left (\int _1^{K[2]}\frac {-K[1]^6+6 K[1]^5-9 K[1]^4+18 K[1]^3-7 K[1]^2+1}{K[1] \left (K[1]^6-3 K[1]^5+K[1]^4-K[1]^2+3 K[1]-1\right )}dK[1]\right ) \, dK[2]\right ) \left (K[2]^2-1\right )}{2 \left (K[2]^2-3 K[2]+1\right )^2}dK[2]-2 c_1 \left (x^2-3 x+1\right )+c_2 (x-1)^2\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**5*Derivative(y(x), (x, 3)) + 4*x**4*Derivative(y(x), (x, 2)) - x**3*Derivative(y(x), (x, 3)) + 12*x**2*y(x) + 2*x**2*Derivative(y(x), (x, 2)) + 4*y(x))/(10*x**3 + 4*x) cannot be solved by the factorable group method