60.2.263 problem 839

Internal problem ID [10850]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 839
Date solved : Monday, January 27, 2025 at 10:15:16 PM
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*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 21

dsolve(diff(y(x),x) = (exp(-y(x)/x)*y(x)+exp(-y(x)/x)*x+x^2)*exp(y(x)/x)/x,y(x), singsol=all)
 

\[ y = \left (\ln \left (2\right )+\ln \left (\frac {x}{-x^{2}+c_{1}}\right )\right ) x \]

✓ Solution by Mathematica

Time used: 0.335 (sec). Leaf size: 533

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

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