54.1.33 problem 33

Internal problem ID [11347]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 33
Date solved : Sunday, October 12, 2025 at 01:35:11 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )}&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 58

ode:=diff(y(x),x)-y(x)^2*diff(f(x),x)/g(x)+diff(g(x),x)/f(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 160

ode=D[y[x],x] - y[x]^2*D[ f[x],x]/g[x] + D[ g[x],x]/f[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

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