60.2.13 problem 589

Internal problem ID [10600]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 589
Date solved : Monday, January 27, 2025 at 09:17:23 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {F \left (-\frac {-1+y \ln \left (x \right )}{y}\right ) y^{2}}{x} \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 51

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

\begin{align*} y &= \frac {1}{\ln \left (x \right )+\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )+1\right )} \\ \int _{\textit {\_b}}^{y}\frac {1}{\left (F \left (\frac {1-\textit {\_a} \ln \left (x \right )}{\textit {\_a}}\right )+1\right ) \textit {\_a}^{2}}d \textit {\_a} -\ln \left (x \right )-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.203 (sec). Leaf size: 245

DSolve[D[y[x],x] == (F[(1 - Log[x]*y[x])/y[x]]*y[x]^2)/x,y[x],x,IncludeSingularSolutions -> True]
 

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