60.2.171 problem 747

Internal problem ID [10758]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 747
Date solved : Monday, January 27, 2025 at 09:41:14 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 62

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

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

✓ Solution by Mathematica

Time used: 0.880 (sec). Leaf size: 89

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

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