60.2.244 problem 820

Internal problem ID [10831]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 820
Date solved : Tuesday, January 28, 2025 at 05:21:14 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Solution by Maple

Time used: 0.109 (sec). Leaf size: 27

dsolve(diff(y(x),x) = 1/2*(-2*cos(y(x))+x^2*cos(2*y(x))*ln(x)+x^2*ln(x))/sin(y(x))/ln(x)/x,y(x), singsol=all)
 

\[ y = \operatorname {arcsec}\left (\frac {2 \ln \left (x \right ) x^{2}-x^{2}+4 c_{1}}{4 \ln \left (x \right )}\right ) \]

✓ Solution by Mathematica

Time used: 1.321 (sec). Leaf size: 77

DSolve[D[y[x],x] == (Csc[y[x]]*(-Cos[y[x]] + (x^2*Log[x])/2 + (x^2*Cos[2*y[x]]*Log[x])/2))/(x*Log[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sec ^{-1}\left (-\frac {x^2-2 x^2 \log (x)+4 c_1}{4 \log (x)}\right ) \\ y(x)\to \sec ^{-1}\left (-\frac {x^2-2 x^2 \log (x)+4 c_1}{4 \log (x)}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}