60.1.151 problem 152

Internal problem ID [10165]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 152
Date solved : Tuesday, January 28, 2025 at 04:26:08 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 142

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)
 

\[ y = \frac {\arctan \left (\frac {6 \sqrt {x^{2}+1}\, \left (\sqrt {x^{2}+1}\, x^{2}+\sqrt {x^{2}+1}+3 c_{1} \right )}{10+6 c_{1} \left (x^{2}+1\right )^{{3}/{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}, \frac {8+6 \left (-x^{2}-1\right ) c_{1} \sqrt {x^{2}+1}-x^{6}-3 x^{4}+6 x^{2}-9 c_{1}^{2}}{10+6 c_{1} \left (x^{2}+1\right )^{{3}/{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}\right )}{2} \]

✓ Solution by Mathematica

Time used: 8.803 (sec). Leaf size: 97

DSolve[(x^2+1)*D[y[x],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 \arctan \left (\frac {x^4+2 x^2-6 c_1 \sqrt {x^2+1}+1}{3 x^2+3}\right ) \\ y(x)\to -\frac {1}{2} \pi \sqrt {\frac {1}{x^2+1}} \sqrt {x^2+1} \\ y(x)\to \frac {1}{2} \pi \sqrt {\frac {1}{x^2+1}} \sqrt {x^2+1} \\ \end{align*}