60.1.539 problem 552
Internal
problem
ID
[10553]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
552
Date
solved
:
Sunday, March 30, 2025 at 06:01:08 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{n}-f \left (x \right ) g \left (y\right )&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 43
ode:=diff(y(x),x)^n-f(x)*g(y(x)) = 0;
dsolve(ode,y(x), singsol=all);
\[
\int _{}^{y}g \left (\textit {\_a} \right )^{-\frac {1}{n}}d \textit {\_a} -g \left (y\right )^{-\frac {1}{n}} \int _{}^{x}\left (f \left (\textit {\_a} \right ) g \left (y\right )\right )^{\frac {1}{n}}d \textit {\_a} +c_1 = 0
\]
✓ Mathematica. Time used: 0.279 (sec). Leaf size: 41
ode=-(f[x]*g[y[x]]) + D[y[x],x]^n==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}g(K[1])^{-1/n}dK[1]\&\right ]\left [\int _1^xf(K[2])^{\frac {1}{n}}dK[2]+c_1\right ]
\]
✓ Sympy. Time used: 3.752 (sec). Leaf size: 76
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
f = Function("f")
g = Function("g")
ode = Eq(-f(x)*g(y(x)) + Derivative(y(x), x)**n,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\begin {cases} g^{- \frac {1}{n}}{\left (y{\left (x \right )} \right )} y{\left (x \right )} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\g^{- \frac {1}{n}}{\left (y{\left (x \right )} \right )} {G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )} + g^{- \frac {1}{n}}{\left (y{\left (x \right )} \right )} {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )} & \text {otherwise} \end {cases} - e^{- \frac {\log {\left (g{\left (y{\left (x \right )} \right )} \right )}}{n}} \int \left (f{\left (x \right )} g{\left (y{\left (x \right )} \right )}\right )^{\frac {1}{n}}\, dx = C_{1}
\]