53.4.40 problem 43

Internal problem ID [8528]
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
Date solved : Monday, January 27, 2025 at 04:09:34 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 4 y {y^{\prime }}^{2} y^{\prime \prime }&={y^{\prime }}^{4}+3 \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 109

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-3\right )^{{3}/{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {4 \left (c_{1} y-3\right )^{{3}/{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {-4 i \left (c_{1} y-3\right )^{{3}/{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {4 i \left (c_{1} y-3\right )^{{3}/{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.252 (sec). Leaf size: 156

DSolve[4*y[x]*(D[y[x],x])^2*D[y[x],{x,2}]==(D[y[x],x])^4+3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (-e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (-i e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (i e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ \end{align*}