29.2.12 problem 37

Internal problem ID [4645]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 2
Problem number : 37
Date solved : Sunday, March 30, 2025 at 03:32:17 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

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

\[ y = -f \left (x \right )-1+{\mathrm e}^{f \left (x \right )} c_1 \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 18

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

\[ y(x)\to -f(x)+c_1 e^{f(x)}-1 \]

✓ Sympy. Time used: 1.210 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
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)
 

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