60.2.416 problem 993

Internal problem ID [11003]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 993
Date solved : Monday, January 27, 2025 at 10:40:06 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 2.234 (sec). Leaf size: 75

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

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