53.4.24 problem 26

Internal problem ID [8512]
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
Date solved : Monday, January 27, 2025 at 04:08:39 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} 2 y^{\prime \prime }&={y^{\prime }}^{3} \sin \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 80

dsolve(2*diff(y(x),x$2)=diff(y(x),x)^3*sin(2*x),y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {-\sin \left (x \right )^{2} c_{1}^{2}+1}\, \operatorname {InverseJacobiAM}\left (x , c_{1}\right )}{\sqrt {\frac {-\sin \left (x \right )^{2} c_{1}^{2}+1}{c_{1}^{2}}}}+c_{2} \\ y &= -\frac {\sqrt {-\sin \left (x \right )^{2} c_{1}^{2}+1}\, \operatorname {InverseJacobiAM}\left (x , c_{1}\right )}{\sqrt {\frac {-\sin \left (x \right )^{2} c_{1}^{2}+1}{c_{1}^{2}}}}+c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.441 (sec). Leaf size: 120

DSolve[2*D[y[x],{x,2}]==(D[y[x],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*}