55.20.9 problem 42

Internal problem ID [13527]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing arbitrary functions and their derivatives.
Problem number : 42
Date solved : Wednesday, October 01, 2025 at 03:59:20 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 44

ode:=diff(y(x),x) = y(x)^2-diff(diff(f(x),x),x)/f(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\int \frac {1}{f \left (x \right )^{2}}d x f^{\prime }\left (x \right ) f \left (x \right )-c_1 f^{\prime }\left (x \right ) f \left (x \right )-1}{\left (\int \frac {1}{f \left (x \right )^{2}}d x +c_1 \right ) f \left (x \right )^{2}} \]

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 132

ode=D[y[x],x]==y[x]^2-D[ f[x],{x,2}]/f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{\left (f(x) K[2]+f''(x)\right )^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2-f''''(K[1])\right )}{\left (f(K[1]) K[2]+f''(K[1])\right )^3}-\frac {2 K[2]}{\left (f(K[1]) K[2]+f''(K[1])\right )^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2-f''''(K[1])}{f(K[1]) \left (f(K[1]) y(x)+f''(K[1])\right )^2}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -y(x)**2 + Derivative(y(x), x) + Derivative(f(x), (x, 2))/f(x) cannot be solved by the lie group method