64.3.12 problem 13

Internal problem ID [13212]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 13
Date solved : Monday, March 31, 2025 at 07:38:09 AM
CAS classification : [_exact, [_Abel, `2nd type`, `class B`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.276 (sec). Leaf size: 24

ode:=2*y(x)*sin(x)*cos(x)+y(x)^2*sin(x)+(sin(x)^2-2*y(x)*cos(x))*diff(y(x),x) = 0; 
ic:=y(0) = 3; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (\sin \left (x \right )^{2}+\sqrt {\sin \left (x \right )^{4}+36 \cos \left (x \right )}\right ) \sec \left (x \right )}{2} \]

✓ Mathematica. Time used: 1.29 (sec). Leaf size: 34

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

\[ y(x)\to \frac {1}{4} \sec (x) \left (-\cos (2 x)+2 \sqrt {\sin ^4(x)+36 \cos (x)}+1\right ) \]

✗ Sympy

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

Timed Out