60.2.200 problem 776

Internal problem ID [10787]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 776
Date solved : Monday, January 27, 2025 at 09:44:22 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 95

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

\[ y = \frac {{\mathrm e}^{-\int \frac {\ln \left (\frac {x^{2}+1}{x}\right ) x +\ln \left (\frac {1}{x}\right )}{x \ln \left (\frac {1}{x}\right )}d x}}{-\int \frac {{\mathrm e}^{-\int \frac {\ln \left (\frac {x^{2}+1}{x}\right ) x +\ln \left (\frac {1}{x}\right )}{x \ln \left (\frac {1}{x}\right )}d x} x \ln \left (\frac {x^{2}+1}{x}\right )}{\ln \left (\frac {1}{x}\right )}d x +c_{1}} \]

✓ Solution by Mathematica

Time used: 0.633 (sec). Leaf size: 110

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

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\left (-\frac {\log \left (K[1]+\frac {1}{K[1]}\right )}{\log \left (\frac {1}{K[1]}\right )}-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[2]}\left (-\frac {\log \left (K[1]+\frac {1}{K[1]}\right )}{\log \left (\frac {1}{K[1]}\right )}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]+\frac {1}{K[2]}\right )}{\log \left (\frac {1}{K[2]}\right )}dK[2]+c_1} \\ y(x)\to 0 \\ \end{align*}