29.4.23 problem 112

Internal problem ID [4714]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 112
Date solved : Tuesday, January 28, 2025 at 02:39:27 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 21

dsolve(diff(y(x),x)+x*(sin(2*y(x))-x^2*cos(y(x))^2) = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \arctan \left (\frac {c_{1} {\mathrm e}^{-x^{2}}}{2}+\frac {x^{2}}{2}-\frac {1}{2}\right ) \]

✓ Solution by Mathematica

Time used: 20.844 (sec). Leaf size: 105

DSolve[D[y[x],x]+x*(Sin[2*y[x]]-x^2*Cos[y[x]]^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \arctan \left (\frac {1}{2} \left (x^2-8 c_1 e^{-x^2}-1\right )\right ) \\ y(x)\to -\arctan \left (-\frac {x^2}{2}+4 c_1 e^{-x^2}+\frac {1}{2}\right ) \\ y(x)\to -\frac {1}{2} \pi e^{x^2} \sqrt {e^{-2 x^2}} \\ y(x)\to \frac {1}{2} \pi e^{x^2} \sqrt {e^{-2 x^2}} \\ \end{align*}