35.8.5 problem 5

Internal problem ID [6212]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page 466
Problem number : 5
Date solved : Monday, January 27, 2025 at 01:47:49 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 75

dsolve(2*x-y(x)*sin(2*x)=(sin(x)^2-2*y(x))*diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y &= \frac {1}{4}-\frac {\cos \left (2 x \right )}{4}-\frac {\sqrt {\cos \left (2 x \right )^{2}-16 x^{2}-2 \cos \left (2 x \right )-16 c_{1} +1}}{4} \\ y &= \frac {1}{4}-\frac {\cos \left (2 x \right )}{4}+\frac {\sqrt {\cos \left (2 x \right )^{2}-16 x^{2}-2 \cos \left (2 x \right )-16 c_{1} +1}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.272 (sec). Leaf size: 89

DSolve[2*x-y[x]*Sin[2*x]==(Sin[x]^2-2*y[x])*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} \left (-\sqrt {-16 x^2+\cos ^2(2 x)-2 \cos (2 x)+1+16 c_1}-\cos (2 x)+1\right ) \\ y(x)\to \frac {1}{4} \left (\sqrt {-16 x^2+\cos ^2(2 x)-2 \cos (2 x)+1+16 c_1}-\cos (2 x)+1\right ) \\ \end{align*}