73.8.22 problem 13.4 (c)

Internal problem ID [15159]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.4 (c)
Date solved : Monday, March 31, 2025 at 01:29:37 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

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

\[ y = \frac {\pi }{2}+\arcsin \left (c_1 x +c_2 \right ) \]

✓ Mathematica. Time used: 0.351 (sec). Leaf size: 81

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sin (K[1])}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sin (K[1])}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sin (K[1])}{c_1}dK[1]\&\right ][x+c_2] \\ \end{align*}

✗ Sympy

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

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