60.2.51 problem 627

Internal problem ID [10638]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 627
Date solved : Monday, January 27, 2025 at 09:19:54 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 35

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

\[ y = \frac {\sin \left (\ln \left (x \right )\right ) c_{1} +\cos \left (\ln \left (x \right )\right )}{\left (\ln \left (x \right )+c_{1} \right ) \cos \left (\ln \left (x \right )\right )+\left (\ln \left (x \right ) c_{1} -1\right ) \sin \left (\ln \left (x \right )\right )} \]

✓ Solution by Mathematica

Time used: 0.901 (sec). Leaf size: 63

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

\begin{align*} y(x)\to \frac {\sin (\log (x))+c_1 \cos (\log (x))}{(1+c_1 \log (x)) \cos (\log (x))+(\log (x)-c_1) \sin (\log (x))} \\ y(x)\to \frac {\cos (\log (x))}{\log (x) \cos (\log (x))-\sin (\log (x))} \\ \end{align*}