54.2.262 problem 840

Internal problem ID [12136]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 840
Date solved : Wednesday, October 01, 2025 at 12:58:05 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{\prime }&=\frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x^{3}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 21

ode:=diff(y(x),x) = (exp(-y(x)/x)*y(x)+exp(-y(x)/x)*x+x^3)*exp(y(x)/x)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\ln \left (3\right )+\ln \left (\frac {x}{-x^{3}+c_1}\right )\right ) x \]

✓ Mathematica. Time used: 0.247 (sec). Leaf size: 387

ode=D[y[x],x] == (E^(y[x]/x)*(x/E^(y[x]/x) + x^3 + y[x]/E^(y[x]/x)))/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {e^{\frac {K[2]}{x}} \int _1^x\left (\frac {e^{\frac {K[2]}{K[1]}} (2 K[1]-K[2])}{3 K[1] \left (e^{\frac {K[2]}{K[1]}} K[1]^2+3\right )}-\frac {e^{\frac {2 K[2]}{K[1]}} K[1] (2 K[1]-K[2])}{3 \left (e^{\frac {K[2]}{K[1]}} K[1]^2+3\right )^2}-\frac {e^{\frac {K[2]}{K[1]}}}{3 \left (e^{\frac {K[2]}{K[1]}} K[1]^2+3\right )}+\frac {1}{3 K[1]^2}\right )dK[1] x^3+3 \int _1^x\left (\frac {e^{\frac {K[2]}{K[1]}} (2 K[1]-K[2])}{3 K[1] \left (e^{\frac {K[2]}{K[1]}} K[1]^2+3\right )}-\frac {e^{\frac {2 K[2]}{K[1]}} K[1] (2 K[1]-K[2])}{3 \left (e^{\frac {K[2]}{K[1]}} K[1]^2+3\right )^2}-\frac {e^{\frac {K[2]}{K[1]}}}{3 \left (e^{\frac {K[2]}{K[1]}} K[1]^2+3\right )}+\frac {1}{3 K[1]^2}\right )dK[1] x+1}{x \left (e^{\frac {K[2]}{x}} x^2+3\right )}dK[2]+\int _1^x\left (\frac {e^{\frac {y(x)}{K[1]}} (2 K[1]-y(x))}{3 \left (e^{\frac {y(x)}{K[1]}} K[1]^2+3\right )}+\frac {y(x)}{3 K[1]^2}+\frac {1}{3 K[1]}\right )dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.764 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**3 + x*exp(-y(x)/x) + y(x)*exp(-y(x)/x))*exp(y(x)/x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + \frac {\log {\left (x \right )}}{3} + \frac {\log {\left (x^{2} e^{\frac {y{\left (x \right )}}{x}} + 3 \right )}}{3} - \frac {y{\left (x \right )}}{3 x} = 0 \]