56.3.26 problem 26

Internal problem ID [8884]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 26
Date solved : Monday, January 27, 2025 at 05:18:28 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 21

dsolve(diff(y(x),x$2)+sin(y(x))*diff(y(x),x)^2=0,y(x), singsol=all)
 

\[ \int _{}^{y}{\mathrm e}^{-\cos \left (\textit {\_a} \right )}d \textit {\_a} -c_{1} x -c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 1.336 (sec). Leaf size: 111

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\sin (K[1])-\cos (K[1]) K[1]}}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{\sin (K[1])-\cos (K[1]) K[1]}}{c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\sin (K[1])-\cos (K[1]) K[1]}}{c_1}dK[1]\&\right ][x+c_2] \\ \end{align*}