60.1.52 problem 53
Internal
problem
ID
[10066]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
53
Date
solved
:
Sunday, March 30, 2025 at 02:59:40 PM
CAS
classification
:
[_Chini, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-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: 0.011 (sec). Leaf size: 215
ode:=diff(y(x),x)-f(x)^(-n+1)*diff(g(x),x)*y(x)^n/((a*g(x)+b)^n)-diff(f(x),x)*y(x)/f(x)-f(x)*diff(g(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\operatorname {RootOf}\left (\left (n a f \left (x \right )^{2-n} {g^{\prime }\left (x \right )}^{3} \left (a g \left (x \right )+b \right )^{-n -1}\right )^{n} f \left (x \right )^{n} \left (a g \left (x \right )+b \right )^{n} \int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_a} f \left (x \right )^{n} \left (a g \left (x \right )+b \right )^{n} \left (n a f \left (x \right )^{2-n} {g^{\prime }\left (x \right )}^{3} \left (a g \left (x \right )+b \right )^{-n -1}\right )^{n}-\textit {\_a}^{n} \left (f \left (x \right )^{1-n} g^{\prime }\left (x \right ) \left (a g \left (x \right )+b \right )^{-n}\right )^{n} \left (f \left (x \right ) g^{\prime }\left (x \right )\right )^{2 n} n^{n}-\left (n a f \left (x \right )^{2-n} {g^{\prime }\left (x \right )}^{3} \left (a g \left (x \right )+b \right )^{-n -1}\right )^{n} f \left (x \right )^{n} \left (a g \left (x \right )+b \right )^{n}}d \textit {\_a} -\ln \left (a g \left (x \right )+b \right )+c_1 \right ) \left (a g \left (x \right )+b \right ) f \left (x \right )}{a}
\]
✓ Mathematica. Time used: 0.907 (sec). Leaf size: 96
ode=D[y[x],x] - f[x]^(1-n)*D[ g[x],x]*y[x]^n/(a*g[x]+b)^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]
\[
\text {Solve}\left [\int _1^{\left (f(x)^{-n} (b+a g(x))^{-n}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (a^n\right )^{\frac {1}{n}} K[1]+1}dK[1]=\frac {f(x) (a g(x)+b) \log (a g(x)+b) \left (f(x)^{-n} (a g(x)+b)^{-n}\right )^{\frac {1}{n}}}{a}+c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
n = symbols("n")
y = Function("y")
f = Function("f")
g = Function("g")
ode = Eq(-f(x)*Derivative(g(x), x) + Derivative(y(x), x) - y(x)*Derivative(f(x), x)/f(x) - f(x)**(1 - n)*y(x)**n*Derivative(g(x), x)/(a*g(x) + b)**n,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded