23.4.157 problem 157

Internal problem ID [6459]
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 : 157
Date solved : Tuesday, September 30, 2025 at 02:57:33 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }&=c y^{2}+b y y^{\prime }+a {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 86

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{-\frac {x \left (b -\sqrt {\left (-4 a +4\right ) c +b^{2}}\right )}{2 a -2}} \left (\frac {-4 c a +b^{2}+4 c}{\left ({\mathrm e}^{x \sqrt {\left (-4 a +4\right ) c +b^{2}}} c_1 -c_2 \right )^{2} \left (a -1\right )^{2}}\right )^{\frac {1}{2 a -2}} \\ \end{align*}

✓ Mathematica. Time used: 11.305 (sec). Leaf size: 51

ode=y[x]*D[y[x],{x,2}] == c*y[x]^2 + b*y[x]*D[y[x],x] + a*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \exp \left (-\frac {b x+2 \log \left (\cos \left (\frac {1}{2} (x+c_1) \sqrt {4 (a-1) c-b^2}\right )\right )}{2 (a-1)}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x) + sqrt((-4*a*c*y(x) + 4*a*Derivat