23.1.47 problem 41

Internal problem ID [4654]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 07:38:17 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

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