60.2.264 problem 840

Internal problem ID [10851]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 840
Date solved : Monday, January 27, 2025 at 10:15:18 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^{3}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 21

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

\[ y = \left (\ln \left (3\right )+\ln \left (\frac {x}{-x^{3}+c_{1}}\right )\right ) x \]

✓ Solution by Mathematica

Time used: 0.403 (sec). Leaf size: 387

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

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