Internal problem ID [7732]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 152.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime }+x \sin \relax (y) \cos \relax (y)-x \left (x^{2}+1\right ) \left (\cos ^{2}\relax (y)\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 191
dsolve((x^2+1)*diff(y(x),x) + x*sin(y(x))*cos(y(x)) - x*(x^2+1)*cos(y(x))^2=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\arctan \left (\frac {6 \sqrt {x^{2}+1}\, x^{4}+12 \sqrt {x^{2}+1}\, x^{2}+18 c_{1} x^{2}+6 \sqrt {x^{2}+1}+18 c_{1}}{\sqrt {x^{2}+1}\, \left (x^{6}+6 \sqrt {x^{2}+1}\, c_{1} x^{2}+3 x^{4}+6 \sqrt {x^{2}+1}\, c_{1}+9 c_{1}^{2}+12 x^{2}+10\right )}, -\frac {x^{6}+6 \sqrt {x^{2}+1}\, c_{1} x^{2}+3 x^{4}+6 \sqrt {x^{2}+1}\, c_{1}+9 c_{1}^{2}-6 x^{2}-8}{x^{6}+6 \sqrt {x^{2}+1}\, c_{1} x^{2}+3 x^{4}+6 \sqrt {x^{2}+1}\, c_{1}+9 c_{1}^{2}+12 x^{2}+10}\right )}{2} \]
✓ Solution by Mathematica
Time used: 6.027 (sec). Leaf size: 110
DSolve[(x^2+1)*y'[x] + x*Sin[y[x]]*Cos[y[x]] - x*(x^2+1)*Cos[y[x]]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \operatorname {ArcTan}\left (\frac {1}{3} \left (x^2-\frac {6 c_1}{\sqrt {x^2+1}}+1\right )\right ) \\ y(x)\to \frac {1}{4} \left (2 i \log \left (-\frac {2 i}{\sqrt {x^2+1}}\right )+i \log \left (\frac {1}{4} \left (x^2+1\right )\right )+\pi \right ) \\ y(x)\to \frac {1}{4} \left (-2 i \log \left (-\frac {i}{\sqrt {x^2+1}}\right )-i \log \left (x^2+1\right )-\pi \right ) \\ \end{align*}