54.1.539 problem 552

Internal problem ID [11853]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 552
Date solved : Friday, October 03, 2025 at 02:48:06 AM
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: 0.183 (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]
 

\begin{align*} 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 ] \end{align*}

✓ Sympy. Time used: 2.416 (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)
 

\[ \begin {cases} g^{- \frac {1}{n}}{\left (y{\left (x \right )} \right )} y{\left (x \right )} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\g^{- \frac {1}{n}}{\left (y{\left (x \right )} \right )} {G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )} + g^{- \frac {1}{n}}{\left (y{\left (x \right )} \right )} {G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )} & \text {otherwise} \end {cases} - e^{- \frac {\log {\left (g{\left (y{\left (x \right )} \right )} \right )}}{n}} \int \left (f{\left (x \right )} g{\left (y{\left (x \right )} \right )}\right )^{\frac {1}{n}}\, dx = C_{1} \]