60.1.510 problem 513

Internal problem ID [10524]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 513
Date solved : Monday, January 27, 2025 at 08:48:58 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} {y^{\prime }}^{2} \sin \left (y\right )+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4}&=0 \end{align*}

✗ Solution by Maple

dsolve(diff(y(x),x)^2*sin(y(x))+2*x*diff(y(x),x)*cos(y(x))^3-sin(y(x))*cos(y(x))^4=0,y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 1.657 (sec). Leaf size: 135

DSolve[-(Cos[y[x]]^4*Sin[y[x]]) + 2*x*Cos[y[x]]^3*D[y[x],x] + Sin[y[x]]*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

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