54.1.84 problem 85

Internal problem ID [11398]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 85
Date solved : Sunday, October 12, 2025 at 01:37:10 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 152

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

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

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 238

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

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

✗ Sympy

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

NotImplementedError : psolve: Cannot solve x*Derivative(y(x, _y), x)/x**a - _y*Derivative(y(x, _y), _y)/_y**b