60.2.223 problem 799

Internal problem ID [10810]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 799
Date solved : Monday, January 27, 2025 at 10:02:46 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\frac {y \left (-1-x \,{\mathrm e}^{\frac {1+x}{x -1}}+x^{2} {\mathrm e}^{\frac {1+x}{x -1}} y-x^{2} {\mathrm e}^{\frac {1+x}{x -1}}+x^{3} {\mathrm e}^{\frac {1+x}{x -1}} y\right )}{x} \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 107

dsolve(diff(y(x),x) = y(x)*(-1-x*exp((x+1)/(x-1))+x^2*exp((x+1)/(x-1))*y(x)-x^2*exp((x+1)/(x-1))+x^3*exp((x+1)/(x-1))*y(x))/x,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\frac {\left (-x^{2}-4 x +5\right ) {\mathrm e}^{\frac {x +1}{x -1}}}{2}-6 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )}}{x \left (c_{1} -\int \left (x +1\right ) {\mathrm e}^{\frac {-\left (x +5\right ) \left (x -1\right )^{2} {\mathrm e}^{\frac {x +1}{x -1}}-12 \left (x -1\right ) {\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )+2 x +2}{2 x -2}}d x \right )} \]

✓ Solution by Mathematica

Time used: 1.258 (sec). Leaf size: 228

DSolve[D[y[x],x] == (y[x]*(-1 - E^((1 + x)/(-1 + x))*x - E^((1 + x)/(-1 + x))*x^2 + E^((1 + x)/(-1 + x))*x^2*y[x] + E^((1 + x)/(-1 + x))*x^3*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\left (-e^{\frac {K[1]+1}{K[1]-1}} (K[1]+1)-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\exp \left (\frac {K[2]+(K[2]-1) \int _1^{K[2]}\left (-e^{\frac {K[1]+1}{K[1]-1}} (K[1]+1)-\frac {1}{K[1]}\right )dK[1]+1}{K[2]-1}\right ) K[2] (K[2]+1)dK[2]+c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {\exp \left (\int _1^x\left (-e^{\frac {K[1]+1}{K[1]-1}} (K[1]+1)-\frac {1}{K[1]}\right )dK[1]\right )}{\int _1^x\exp \left (\frac {K[2]+(K[2]-1) \int _1^{K[2]}\left (-e^{\frac {K[1]+1}{K[1]-1}} (K[1]+1)-\frac {1}{K[1]}\right )dK[1]+1}{K[2]-1}\right ) K[2] (K[2]+1)dK[2]} \\ \end{align*}