23.4.275 problem 275

Internal problem ID [6577]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 275
Date solved : Tuesday, September 30, 2025 at 03:07:56 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \operatorname {a2} \left (\operatorname {a3} +\operatorname {a1} \sin \left (y\right )^{2}\right ) y+\operatorname {a1} {y^{\prime }}^{2}+\operatorname {a1} \cos \left (y\right ) \sin \left (y\right ) {y^{\prime }}^{2}+\left (\operatorname {a0} +\operatorname {a1} \sin \left (y\right )^{2}\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 687

ode:=a2*(a3+a1*sin(y(x))^2)*y(x)+a1*diff(y(x),x)^2+a1*cos(y(x))*sin(y(x))*diff(y(x),x)^2+(a0+a1*sin(y(x))^2)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 6.83 (sec). Leaf size: 1154

ode=a2*(a3 + a1*Sin[y[x]]^2)*y[x] + a1*D[y[x],x]^2 + a1*Cos[y[x]]*Sin[y[x]]*D[y[x],x]^2 + (a0 + a1*Sin[y[x]]^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(a1*sin(y(x))*cos(y(x))*Derivative(y(x), x)**2 + a1*Derivative(y(x), x)**2 + a2*(a1*sin(y(x))**2 + a3)*y(x) + (a0 + a1*sin(y(x))**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-(a0*Derivative(y(x), (x, 2)) + a1*a2*y(x)*sin(y(x))**2 +