23.1.132 problem 136

Internal problem ID [4739]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 136
Date solved : Sunday, October 12, 2025 at 01:18:10 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right ) \end{align*}

✓ Maple. Time used: 0.289 (sec). Leaf size: 174

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

\[ y = \left (-\frac {a \,x^{m}-\operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (\left (\frac {\textit {\_a} b -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {\textit {\_a} b -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (m^{\frac {1}{m}}\right )^{m} b n \textit {\_a} -\left (\left (\frac {\textit {\_a} b -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {\textit {\_a} b -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (m^{\frac {1}{m}}\right )^{m} a m n +a \,m^{2}}d \textit {\_a} b \,m^{2}+c_1 m -x^{m}\right ) b}{b}\right )^{\frac {1}{n}} \]

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 242

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

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

✗ Sympy

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

Timed Out