58.2.2 problem 2

Internal problem ID [9125]
Book : Second order enumerated odes
Section : section 2
Problem number : 2
Date solved : Monday, January 27, 2025 at 05:45:47 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 35

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

\[ y = -i \operatorname {RootOf}\left (i \sqrt {2}\, c_{1} \left (\int {\mathrm e}^{\cos \left (x \right )}d x \right )+i \sqrt {2}\, c_{2} -\operatorname {erf}\left (\textit {\_Z} \right ) \sqrt {\pi }\right ) \sqrt {2} \]

✓ Solution by Mathematica

Time used: 66.642 (sec). Leaf size: 76

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

\begin{align*} y(x)\to -i \sqrt {2} \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} \left (\int _1^x-e^{\cos (K[2])} c_1dK[2]+c_2\right )\right ) \\ y(x)\to -i \sqrt {2} \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} c_2\right ) \\ \end{align*}