60.2.53 problem 629

Internal problem ID [10640]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 629
Date solved : Monday, January 27, 2025 at 09:20:00 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 62

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

\[ y = \frac {\sin \left (\sqrt {2}\, \ln \left (x \right )\right ) c_{1} +\cos \left (\sqrt {2}\, \ln \left (x \right )\right )}{\sin \left (\sqrt {2}\, \ln \left (x \right )\right ) \left (2 \ln \left (x \right ) c_{1} -\sqrt {2}\right )+\left (c_{1} \sqrt {2}+2 \ln \left (x \right )\right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )} \]

✓ Solution by Mathematica

Time used: 0.935 (sec). Leaf size: 123

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

\begin{align*} y(x)\to \frac {\sin \left (\sqrt {2} \log (x)\right )+c_1 \cos \left (\sqrt {2} \log (x)\right )}{\left (\sqrt {2}+2 c_1 \log (x)\right ) \cos \left (\sqrt {2} \log (x)\right )+\left (2 \log (x)-\sqrt {2} c_1\right ) \sin \left (\sqrt {2} \log (x)\right )} \\ y(x)\to \frac {\cos \left (\sqrt {2} \log (x)\right )}{2 \log (x) \cos \left (\sqrt {2} \log (x)\right )-\sqrt {2} \sin \left (\sqrt {2} \log (x)\right )} \\ \end{align*}