60.2.166 problem 742
Internal
problem
ID
[10753]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
742
Date
solved
:
Tuesday, January 28, 2025 at 05:10:42 PM
CAS
classification
:
unknown
\begin{align*} y^{\prime }&=-\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (x \sin \left (y\right )-1\right ) \left (1+x \right )} \end{align*}
✓ Solution by Maple
Time used: 0.743 (sec). Leaf size: 239
dsolve(diff(y(x),x) = -cos(y(x))/(x*sin(y(x))-1)*(x-cos(y(x))+1)/(x+1),y(x), singsol=all)
\begin{align*}
y &= \arctan \left (\frac {\left (-\ln \left (x +1\right )+c_{1} \right ) \sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}+x}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}, \frac {\ln \left (x +1\right ) x -c_{1} x +\sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}\right ) \\
y &= \arctan \left (\frac {\left (\ln \left (x +1\right )-c_{1} \right ) \sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}+x}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}, \frac {\ln \left (x +1\right ) x -c_{1} x -\sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}\right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 61.801 (sec). Leaf size: 221
DSolve[D[y[x],x] == -(((1 + x - Cos[y[x]])*Cos[y[x]])/((1 + x)*(-1 + x*Sin[y[x]]))),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sec ^{-1}\left (\frac {-\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\
y(x)\to \sec ^{-1}\left (\frac {-\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\
y(x)\to -\sec ^{-1}\left (\frac {\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\
y(x)\to \sec ^{-1}\left (\frac {\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\
\end{align*}