78.7.2 problem 1 (b)

Internal problem ID [18188]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 11 (Reduction of order). Problems at page 87
Problem number : 1 (b)
Date solved : Tuesday, January 28, 2025 at 11:37:34 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 31

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

\begin{align*} y &= -\sqrt {-x^{2}+c_1}+c_2 \\ y &= \sqrt {-x^{2}+c_1}+c_2 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.359 (sec). Leaf size: 103

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

\begin{align*} y(x)\to c_2-i e^{-c_1} \sqrt {-1+e^{2 c_1} x^2} \\ y(x)\to i e^{-c_1} \sqrt {-1+e^{2 c_1} x^2}+c_2 \\ y(x)\to c_2-i \sqrt {x^2} \\ y(x)\to i \sqrt {x^2}+c_2 \\ \end{align*}