23.4.198 problem 198

Internal problem ID [6500]
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 : 198
Date solved : Tuesday, September 30, 2025 at 03:02:22 PM
CAS classification : [NONE]

\begin{align*} a \left (2+a \right )^{2} y y^{\prime \prime }&=-a^{2} f \left (x \right )^{2} y^{4}+a^{2} \left (2+a \right ) y^{3} f^{\prime }\left (x \right )+a \left (2+a \right )^{2} f \left (x \right ) y^{2} y^{\prime }+\left (-1+a \right ) \left (2+a \right )^{2} {y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=a*(2+a)^2*y(x)*diff(diff(y(x),x),x) = -a^2*f(x)^2*y(x)^4+a^2*(2+a)*y(x)^3*diff(f(x),x)+a*(2+a)^2*f(x)*y(x)^2*diff(y(x),x)+(a-1)*(2+a)^2*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 25.351 (sec). Leaf size: 46

ode=a*(2 + a)^2*y[x]*D[y[x],{x,2}] == -(a^2*f[x]^2*y[x]^4) + a^2*(2 + a)*y[x]^3*D[f[x],x] + a*(2 + a)^2*f[x]*y[x]^2*D[y[x],x] + (-1 + a)*(2 + a)^2*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {(a+2) (x+c_1){}^a}{a \int _1^xf(K[5]) (c_1+K[5]){}^adK[5]+c_2}\\ y(x)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*(a + 2)*y(x)**3*Derivative(f(x), x) + a**2*f(x)**2*y(x)**4 - a*(a + 2)**2*f(x)*y(x)**2*Derivative(y(x), x) + a*(a + 2)**2*y(x)*Derivative(y(x), (x, 2)) - (a - 1)*(a + 2)**2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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