60.4.71 problem 1529

Internal problem ID [11494]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1529
Date solved : Sunday, March 30, 2025 at 08:23:35 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

ode:=(sin(x)+x)*diff(diff(diff(y(x),x),x),x)+3*(cos(x)+1)*diff(diff(y(x),x),x)-3*diff(y(x),x)*sin(x)-y(x)*cos(x)+sin(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_3 +c_1 \,x^{2}+c_2 x -\cos \left (x \right )}{\sin \left (x \right )+x} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 76

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

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

✗ Sympy

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

NotImplementedError : The given ODE -(x*Derivative(y(x), (x, 3)) + (Derivative(y(x), (x, 3)) + 1)*sin(x) - y(x)*cos(x) + 3*cos(x)*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), (x, 2)))/(3*sin(x)) + Derivative(y(x), x) cannot be solved by the factorable group method