Internal problem ID [6107]
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: 43.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {4 y \left (y^{\prime }\right )^{2} y^{\prime \prime }-\left (y^{\prime }\right )^{4}-3=0} \end {gather*}
✓ Solution by Maple
Time used: 0.169 (sec). Leaf size: 91
dsolve(4*y(x)*diff(y(x),x)^2*diff(y(x),x$2)=diff(y(x),x)^4+3,y(x), singsol=all)
\begin{align*} -\frac {4 \left (c_{1} y \relax (x )-3\right )^{\frac {3}{4}}}{3 c_{1}}-x -c_{2} = 0 \\ \frac {4 \left (c_{1} y \relax (x )-3\right )^{\frac {3}{4}}}{3 c_{1}}-x -c_{2} = 0 \\ -\frac {4 i \left (c_{1} y \relax (x )-3\right )^{\frac {3}{4}}}{3 c_{1}}-x -c_{2} = 0 \\ \frac {4 i \left (c_{1} y \relax (x )-3\right )^{\frac {3}{4}}}{3 c_{1}}-x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.235 (sec). Leaf size: 140
DSolve[4*y[x]*(y'[x])^2*y''[x]==(y'[x])^4+3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3 \left (8+\sqrt [3]{6} \left (-c_1{}^4 (x+c_2)\right ){}^{4/3}\right )}{8 c_1{}^4} \\ y(x)\to \frac {3 \left (8+\sqrt [3]{6} \left (-i c_1{}^4 (x+c_2)\right ){}^{4/3}\right )}{8 c_1{}^4} \\ y(x)\to \frac {3 \left (8+\sqrt [3]{6} \left (i c_1{}^4 (x+c_2)\right ){}^{4/3}\right )}{8 c_1{}^4} \\ y(x)\to \frac {3 \left (8+\sqrt [3]{6} \left (c_1{}^4 (x+c_2)\right ){}^{4/3}\right )}{8 c_1{}^4} \\ \end{align*}