53.4.28 problem 31

Internal problem ID [8516]
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
Date solved : Monday, January 27, 2025 at 04:08:51 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y y^{\prime \prime }&={y^{\prime }}^{2} \left (1-y^{\prime } \sin \left (y\right )-y y^{\prime } \cos \left (y\right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.172 (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 &= c_{1} \\ -\cos \left (y\right )+c_{1} \ln \left (y\right )-x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.456 (sec). Leaf size: 69

DSolve[y[x]*D[y[x],{x,2}]==(D[y[x],x])^2*(1-D[y[x],x]*Sin[y[x]]-y[x]*D[y[x],x]*Cos[y[x]] ),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}[-\cos (\text {$\#$1})+c_1 \log (\text {$\#$1})\&][x+c_2] \\ y(x)\to \text {InverseFunction}[-\cos (\text {$\#$1})-c_1 \log (\text {$\#$1})\&][x+c_2] \\ y(x)\to \text {InverseFunction}[-\cos (\text {$\#$1})+c_1 \log (\text {$\#$1})\&][x+c_2] \\ \end{align*}