Internal problem ID [10924]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 19(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y y^{\prime \prime } \sin \left (x \right )+\left (y \cos \left (x \right )+\sin \left (x \right ) y^{\prime }\right ) y^{\prime }-\cos \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 119
dsolve(y(x)*diff(y(x),x$2)*sin(x)+ ( diff(y(x),x)*sin(x)+y(x)*cos(x) )*diff(y(x),x)=cos(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \sqrt {\sqrt {2}\, \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right ) c_{2} -\sqrt {2}\, \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (\cos \left (x \right )\right ) c_{1} +2 \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \left (\int \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (\cos \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right ) \cos \left (x \right )d x \right ) \operatorname {csgn}\left (\cos \left (x \right )\right )-2 \,\operatorname {arctanh}\left (\cos \left (x \right )\right ) \sin \left (x \right )} \\ y \left (x \right ) = -\sqrt {\sqrt {2}\, \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right ) c_{2} -\sqrt {2}\, \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (\cos \left (x \right )\right ) c_{1} +2 \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \left (\int \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (\cos \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right ) \cos \left (x \right )d x \right ) \operatorname {csgn}\left (\cos \left (x \right )\right )-2 \,\operatorname {arctanh}\left (\cos \left (x \right )\right ) \sin \left (x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 78
DSolve[y[x]*y''[x]*Sin[x]+ ( y'[x]*Sin[x]+y[x]*Cos[x] )*y'[x]==Cos[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {x+c_1 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+c_2} \\ y(x)\to \sqrt {2} \sqrt {x+c_1 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+c_2} \\ \end{align*}