64.3.12 problem 13

Internal problem ID [13283]
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 : Tuesday, January 28, 2025 at 05:14:29 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*}

✓ Solution by Maple

Time used: 0.423 (sec). Leaf size: 24

dsolve([(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,y(0) = 3],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.290 (sec). Leaf size: 34

DSolve[{(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,{y[0]==3}},y[x],x,IncludeSingularSolutions -> True]
 

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