60.2.285 problem 861

Internal problem ID [10872]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 861
Date solved : Tuesday, January 28, 2025 at 05:27:31 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x} \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 39

dsolve(diff(y(x),x) = -(-1/x*y(x)/exp(-1/x)-_F1(y(x)/exp(-1/x)))*exp(-1/x)/x,y(x), singsol=all)
 

\begin{align*} y &= \operatorname {RootOf}\left (f_{1} \left (\textit {\_Z} \right )\right ) {\mathrm e}^{-\frac {1}{x}} \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{-\frac {1}{x}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.644 (sec). Leaf size: 152

DSolve[D[y[x],x] == (F1[E^x^(-1)*y[x]] + (E^x^(-1)*y[x])/x)/(E^x^(-1)*x),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}\left (e^{\frac {1}{x}} K[2]\right ) \int _1^x\left (\frac {e^{\frac {1}{K[1]}}}{\text {F1}\left (e^{\frac {1}{K[1]}} K[2]\right ) K[1]^2}-\frac {e^{\frac {2}{K[1]}} K[2] \text {F1}''\left (e^{\frac {1}{K[1]}} K[2]\right )}{\text {F1}\left (e^{\frac {1}{K[1]}} K[2]\right )^2 K[1]^2}\right )dK[1]+e^{\frac {1}{x}}}{\text {F1}\left (e^{\frac {1}{x}} K[2]\right )}dK[2]+\int _1^x\left (\frac {e^{\frac {1}{K[1]}} y(x)}{\text {F1}\left (e^{\frac {1}{K[1]}} y(x)\right ) K[1]^2}+\frac {1}{K[1]}\right )dK[1]=c_1,y(x)\right ] \]