60.2.169 problem 745

Internal problem ID [10756]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 745
Date solved : Monday, January 27, 2025 at 09:41:09 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], [_Abel, `2nd type`, `class C`]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 77

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

\[ y = \frac {47 \operatorname {RootOf}\left (-27783 \left (\int _{}^{\textit {\_Z}}\frac {1}{2209 \textit {\_a}^{3}-9261 \textit {\_a} +9261}d \textit {\_a} \right )-7 \ln \left (x \right )+3 c_{1} \right )-84}{21+47 \left (-1+\ln \left (x \right )\right ) \operatorname {RootOf}\left (-27783 \left (\int _{}^{\textit {\_Z}}\frac {1}{2209 \textit {\_a}^{3}-9261 \textit {\_a} +9261}d \textit {\_a} \right )-7 \ln \left (x \right )+3 c_{1} \right )-84 \ln \left (x \right )} \]

✓ Solution by Mathematica

Time used: 0.549 (sec). Leaf size: 109

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

\[ \text {Solve}\left [\int _1^{\frac {-4 \log (x) y(x)+y(x)+4}{\sqrt [3]{47} \sqrt [3]{-\frac {1}{(\log (x)-1)^3}} (\log (x)-1) ((\log (x)-1) y(x)-1)}}\frac {1}{K[1]^3+\frac {21 \sqrt [3]{-1} K[1]}{47^{2/3}}+1}dK[1]=\frac {1}{9} 47^{2/3} \left (-\frac {1}{(\log (x)-1)^3}\right )^{5/3} \log (x) (\log (x)-1)^5+c_1,y(x)\right ] \]