23.4.33 problem 33

Internal problem ID [6335]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 33
Date solved : Friday, October 03, 2025 at 02:01:33 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=a \left (1+2 y y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 58

ode:=diff(diff(y(x),x),x) = a*(1+2*y(x)*diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\[ \int _{}^{y}\frac {1}{\textit {\_a}^{2} a +\operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) \left (-a \right )^{{1}/{3}} c_1 \textit {\_a} +\operatorname {AiryAi}\left (\textit {\_Z} \right ) \left (-a \right )^{{1}/{3}} \textit {\_a} +\operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_1 +\operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right ) \left (-a \right )^{{1}/{3}}}d \textit {\_a} -x -c_2 = 0 \]

✓ Mathematica. Time used: 120.157 (sec). Leaf size: 102

ode=D[y[x],{x,2}] == a*(1 + 2*y[x]*D[y[x],x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {a \left (c_2 \operatorname {AiryAiPrime}\left (\frac {a (c_1-a x)}{\left (-a^2\right )^{2/3}}\right )+\operatorname {AiryBiPrime}\left (\frac {a (c_1-a x)}{\left (-a^2\right )^{2/3}}\right )\right )}{\left (-a^2\right )^{2/3} \left (c_2 \operatorname {AiryAi}\left (\frac {a (c_1-a x)}{\left (-a^2\right )^{2/3}}\right )+\operatorname {AiryBi}\left (\frac {a (c_1-a x)}{\left (-a^2\right )^{2/3}}\right )\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(2*y(x)*Derivative(y(x), x) + 1) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a + Derivative(y(x), (x, 2)))/(2*a*y(x))