35.7.7 problem 4

Internal problem ID [6189]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 4
Date solved : Monday, January 27, 2025 at 01:47:09 PM
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: 0.017 (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: 1.740 (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*}