60.2.324 problem 901

Internal problem ID [10911]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 901
Date solved : Monday, January 27, 2025 at 10:23:29 PM
CAS classification : [NONE]

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

✓ Solution by Maple

Time used: 0.495 (sec). Leaf size: 31

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (2 a x \textit {\_Z} -2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-x^{2}+2 c_{1} \right )} \]

✓ Solution by Mathematica

Time used: 0.405 (sec). Leaf size: 80

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

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