Internal problem ID [6091]
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: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime }-\left (y^{\prime }\right )^{3} \sin \left (2 x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 158
dsolve(2*diff(y(x),x$2)=diff(y(x),x)^3*sin(2*x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c_{2} \sin \relax (x ) \sqrt {4 c_{1}-2+2 \cos \left (2 x \right )}-\EllipticF \left (\cos \relax (x ), \sqrt {-\frac {1}{c_{1}-1}}\right ) \sqrt {\frac {2 c_{1}-1+\cos \left (2 x \right )}{c_{1}-1}}\, \sqrt {1-\cos \left (2 x \right )}}{2 \sin \relax (x ) \sqrt {c_{1}-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}}} \\ y \relax (x ) = \frac {\EllipticF \left (\cos \relax (x ), \sqrt {-\frac {1}{c_{1}-1}}\right ) \sqrt {\frac {2 c_{1}-1+\cos \left (2 x \right )}{c_{1}-1}}\, \sqrt {1-\cos \left (2 x \right )}+c_{2} \sin \relax (x ) \sqrt {4 c_{1}-2+2 \cos \left (2 x \right )}}{2 \sin \relax (x ) \sqrt {c_{1}-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 9.324 (sec). Leaf size: 118
DSolve[2*y''[x]==(y'[x])^3*Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {\sqrt {\frac {\cos (2 x)+1-4 c_1}{1-2 c_1}} \text {EllipticF}\left (x,\frac {1}{1-2 c_1}\right )}{\sqrt {\cos (2 x)+1-4 c_1}} \\ y(x)\to \frac {\sqrt {\frac {\cos (2 x)+1-4 c_1}{1-2 c_1}} \text {EllipticF}\left (x,\frac {1}{1-2 c_1}\right )}{\sqrt {\cos (2 x)+1-4 c_1}}+c_2 \\ y(x)\to c_2 \\ \end{align*}