60.1.10 problem 10

Internal problem ID [10024]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 08:04:37 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.000 (sec). Leaf size: 15

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

\[ y = f-1+{\mathrm e}^{-f} c_{1} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 60

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

\[ y(x)\to \exp \left (\int _1^x-f''(K[1])dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}-f''(K[1])dK[1]\right ) f(K[2]) f''(K[2])dK[2]+c_1\right ) \]

✓ Sympy. Time used: 0.966 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{1} e^{- f{\left (x \right )}} + f{\left (x \right )} - 1 \]