29.7.7 problem 4

Internal problem ID [7324]
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 : Tuesday, September 30, 2025 at 04:28:53 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*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 31

ode:=x*diff(diff(y(x),x),x) = diff(y(x),x)+diff(y(x),x)^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.449 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (K[1]^2+1\right )}dK[1]\&\right ][c_1+\log (K[2])]dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 9.501 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**3 - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - x^{2} \sqrt {- \frac {C_{2}}{C_{2} x^{2} - 1}} + \frac {\sqrt {- \frac {C_{2}}{C_{2} x^{2} - 1}}}{C_{2}}, \ y{\left (x \right )} = C_{1} + x^{2} \sqrt {- \frac {C_{2}}{C_{2} x^{2} - 1}} - \frac {\sqrt {- \frac {C_{2}}{C_{2} x^{2} - 1}}}{C_{2}}\right ] \]