53.4.24 problem 26

Internal problem ID [8512]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 26
Date solved : Sunday, March 30, 2025 at 01:12:48 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} 2 y^{\prime \prime }&={y^{\prime }}^{3} \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 80

ode:=2*diff(diff(y(x),x),x) = diff(y(x),x)^3*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {-\sin \left (x \right )^{2} c_1^{2}+1}\, \operatorname {InverseJacobiAM}\left (x , c_1\right )}{\sqrt {\frac {-\sin \left (x \right )^{2} c_1^{2}+1}{c_1^{2}}}}+c_2 \\ y &= -\frac {\sqrt {-\sin \left (x \right )^{2} c_1^{2}+1}\, \operatorname {InverseJacobiAM}\left (x , c_1\right )}{\sqrt {\frac {-\sin \left (x \right )^{2} c_1^{2}+1}{c_1^{2}}}}+c_2 \\ \end{align*}

✓ Mathematica. Time used: 5.441 (sec). Leaf size: 120

ode=2*D[y[x],{x,2}]==(D[y[x],x])^3*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_2-\frac {\sqrt {-\frac {\cos (2 x)+1-4 c_1}{-1+2 c_1}} \operatorname {EllipticF}\left (x,\frac {1}{1-2 c_1}\right )}{\sqrt {\cos (2 x)+1-4 c_1}} \\ y(x)\to \frac {\sqrt {-\frac {\cos (2 x)+1-4 c_1}{-1+2 c_1}} \operatorname {EllipticF}\left (x,\frac {1}{1-2 c_1}\right )}{\sqrt {\cos (2 x)+1-4 c_1}}+c_2 \\ y(x)\to c_2 \\ \end{align*}

✗ Sympy

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

Timed Out