60.1.84 problem 85
Internal
problem
ID
[10098]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
85
Date
solved
:
Sunday, March 30, 2025 at 03:15:51 PM
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: 0.33 (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