Internal problem ID [6095]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES
Page 324
Problem number: 31.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {y y^{\prime \prime }-\left (y^{\prime }\right )^{2} \left (1-y^{\prime } \sin \relax (y)-y y^{\prime } \cos \relax (y)\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.485 (sec). Leaf size: 24
dsolve(y(x)*diff(y(x),x$2)=diff(y(x),x)^2*(1-diff(y(x),x)*sin(y(x))-y(x)*diff(y(x),x)*cos(y(x)) ),y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} \\ -\cos \left (y \relax (x )\right )+c_{1} \ln \left (y \relax (x )\right )-x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.258 (sec). Leaf size: 23
DSolve[y[x]*y''[x]==(y'[x])^2*(1-y'[x]*Sin[y[x]]-y[x]*y'[x]*Cos[y[x]] ),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}[-\cos (\text {$\#$1})+c_1 \log (\text {$\#$1})\&][x+c_2] \\ \end{align*}