60.4.17 problem 1470

Internal problem ID [11440]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1470
Date solved : Sunday, March 30, 2025 at 08:22:16 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 40

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)*sin(x)-2*diff(y(x),x)*cos(x)+y(x)*sin(x)-ln(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_3 +\frac {\int \left (8 c_1 x +4 c_2 -3 x^{2}+2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{\cos \left (x \right )}d x}{4}\right ) {\mathrm e}^{-\cos \left (x \right )} \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 73

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

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

✗ Sympy

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

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