60.4.70 problem 1528

Internal problem ID [11493]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1528
Date solved : Sunday, March 30, 2025 at 08:23:31 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 55

ode:=diff(diff(diff(y(x),x),x),x)*sin(x)+(2*cos(x)+1)*diff(diff(y(x),x),x)-diff(y(x),x)*sin(x)-cos(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 1-2 c_1 \left (-x +\left (\csc \left (x \right )+\cot \left (x \right )\right ) \left (\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-\ln \left (\sin \left (x \right )\right )\right )\right )+c_2 +\frac {c_3}{\csc \left (x \right )-\cot \left (x \right )}-x \cot \left (x \right )-x \csc \left (x \right ) \]

✓ Mathematica. Time used: 19.821 (sec). Leaf size: 233

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

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

✗ Sympy

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

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