Internal problem ID [6090]
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]]
\[ \boxed {2 y^{\prime \prime }-{y^{\prime }}^{3} \sin \left (2 x \right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 78
dsolve(2*diff(y(x),x$2)=diff(y(x),x)^3*sin(2*x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\sqrt {-\left (\sin \left (x \right )^{2}-\frac {1}{c_{1}^{2}}\right ) c_{1}^{2}}\, \operatorname {InverseJacobiAM}\left (x , c_{1}\right )}{\sqrt {-\sin \left (x \right )^{2}+\frac {1}{c_{1}^{2}}}}+c_{2} \\ y \left (x \right ) = -\frac {\sqrt {-\left (\sin \left (x \right )^{2}-\frac {1}{c_{1}^{2}}\right ) c_{1}^{2}}\, \operatorname {InverseJacobiAM}\left (x , c_{1}\right )}{\sqrt {-\sin \left (x \right )^{2}+\frac {1}{c_{1}^{2}}}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.916 (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}} \operatorname {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}} \operatorname {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*}