60.1.53 problem 54

Internal problem ID [10067]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 54
Date solved : Sunday, March 30, 2025 at 03:00:26 PM
CAS classification : [_Chini, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 38

ode:=diff(y(x),x)-a^n*f(x)^(-n+1)*diff(g(x),x)*y(x)^n-diff(f(x),x)*y(x)/f(x)-f(x)*diff(g(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {a y \operatorname {LerchPhi}\left (-\left (\frac {a y}{f \left (x \right )}\right )^{n}, 1, \frac {1}{n}\right )}{n f \left (x \right )}-a g \left (x \right )+c_1 = 0 \]

✓ Mathematica. Time used: 0.338 (sec). Leaf size: 86

ode=D[y[x],x] - a^n*f[x]^(1-n)*D[ g[x],x]*y[x]^n - D[ f[x],x]*y[x]/f[x] - f[x]*D[ g[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [y(x) \left (a^n f(x)^{-n}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\left (\left (a^n f(x)^{-n}\right )^{\frac {1}{n}} y(x)\right )^n\right )=\int _1^xf(K[1]) \left (a^n f(K[1])^{-n}\right )^{\frac {1}{n}} g''(K[1])dK[1]+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
f = Function("f") 
g = Function("g") 
ode = Eq(-a**n*f(x)**(1 - n)*y(x)**n*Derivative(g(x), x) - f(x)*Derivative(g(x), x) + Derivative(y(x), x) - y(x)*Derivative(f(x), x)/f(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out