53.4.1 problem 1

Internal problem ID [8489]
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 : 1
Date solved : Monday, January 27, 2025 at 04:07:17 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=x {y^{\prime }}^{3} \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 10.839 (sec). Leaf size: 57

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

\begin{align*} y(x)\to \arctan \left (\frac {x}{\sqrt {-x^2-2 c_1}}\right )+c_2 \\ y(x)\to c_2-\arctan \left (\frac {x}{\sqrt {-x^2-2 c_1}}\right ) \\ y(x)\to c_2 \\ \end{align*}