54.2.261 problem 839

Internal problem ID [12135]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 839
Date solved : Wednesday, October 01, 2025 at 12:58:03 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^{2}\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^2)*exp(y(x)/x)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\ln \left (2\right )+\ln \left (\frac {x}{-x^{2}+c_1}\right )\right ) x \]

✓ Mathematica. Time used: 0.207 (sec). Leaf size: 533

ode=D[y[x],x] == (E^(y[x]/x)*(x/E^(y[x]/x) + x^2 + y[x]/E^(y[x]/x)))/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {e^{\frac {K[2]}{x}} \int _1^x\left (\frac {e^{\frac {K[2]}{K[1]}} \left (e^{\frac {K[2]}{K[1]}} K[2]+2\right )}{4 K[1] \left (e^{\frac {K[2]}{K[1]}} K[1]+2\right )}-\frac {e^{\frac {2 K[2]}{K[1]}} \left (e^{\frac {K[2]}{K[1]}} K[2]+2\right )}{4 \left (e^{\frac {K[2]}{K[1]}} K[1]+2\right )^2}+\frac {-\frac {e^{\frac {K[2]}{K[1]}} K[2]}{K[1]}-e^{\frac {K[2]}{K[1]}}}{4 K[1]}+\frac {e^{\frac {K[2]}{K[1]}} \left (\frac {e^{\frac {K[2]}{K[1]}} K[2]}{K[1]}+e^{\frac {K[2]}{K[1]}}\right )}{4 \left (e^{\frac {K[2]}{K[1]}} K[1]+2\right )}+\frac {1}{2 K[1]^2}\right )dK[1] x^2+2 \int _1^x\left (\frac {e^{\frac {K[2]}{K[1]}} \left (e^{\frac {K[2]}{K[1]}} K[2]+2\right )}{4 K[1] \left (e^{\frac {K[2]}{K[1]}} K[1]+2\right )}-\frac {e^{\frac {2 K[2]}{K[1]}} \left (e^{\frac {K[2]}{K[1]}} K[2]+2\right )}{4 \left (e^{\frac {K[2]}{K[1]}} K[1]+2\right )^2}+\frac {-\frac {e^{\frac {K[2]}{K[1]}} K[2]}{K[1]}-e^{\frac {K[2]}{K[1]}}}{4 K[1]}+\frac {e^{\frac {K[2]}{K[1]}} \left (\frac {e^{\frac {K[2]}{K[1]}} K[2]}{K[1]}+e^{\frac {K[2]}{K[1]}}\right )}{4 \left (e^{\frac {K[2]}{K[1]}} K[1]+2\right )}+\frac {1}{2 K[1]^2}\right )dK[1] x+1}{x \left (e^{\frac {K[2]}{x}} x+2\right )}dK[2]+\int _1^x\left (\frac {y(x)}{2 K[1]^2}+\frac {2-e^{\frac {y(x)}{K[1]}} y(x)}{4 K[1]}+\frac {e^{\frac {y(x)}{K[1]}} \left (e^{\frac {y(x)}{K[1]}} y(x)+2\right )}{4 \left (e^{\frac {y(x)}{K[1]}} K[1]+2\right )}\right )dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.447 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**2 + 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)
 

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