60.1.558 problem 571

Internal problem ID [10572]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 571
Date solved : Sunday, March 30, 2025 at 06:08:02 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.545 (sec). Leaf size: 169

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

\[ \left [y \left (\textit {\_T} \right ) = a {\left (\left (\frac {\left (-n +1\right ) \int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} +c_1 a n}{f \left (\textit {\_T} \right ) a n}\right )^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}\right )}^{n} f \left (\textit {\_T} \right )+\left (\frac {\left (-n +1\right ) \int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} +c_1 a n}{f \left (\textit {\_T} \right ) a n}\right )^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}} \textit {\_T}, x \left (\textit {\_T} \right ) = \left (\frac {\left (-n +1\right ) \int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} +c_1 a n}{f \left (\textit {\_T} \right ) a n}\right )^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}\right ] \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 124

ode=a*x^n*f[D[y[x],x]] - y[x] + x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : multiple generators [_X0, f(_X0)] 
No algorithms are implemented to solve equation _X0*x + a*x**n*f(_X0) - y(x)