61.8.5 problem 14

Internal problem ID [12165]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 12:46:07 AM
CAS classification : [_linear]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 51

dsolve(diff(y(x),x)=a*(ln(x))^n*y(x)-a*b*x*(ln(x))^(n+1)*y(x)+b*ln(x)+b,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.225 (sec). Leaf size: 78

DSolve[D[y[x],x]==a*(Log[x])^n*y[x]-a*b*x*(Log[x])^(n+1)*y[x]+b*Log[x]+b,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-a \log ^n(K[1]) (b K[1] \log (K[1])-1)dK[1]\right ) \left (\int _1^xb \exp \left (-\int _1^{K[2]}-a \log ^n(K[1]) (b K[1] \log (K[1])-1)dK[1]\right ) (\log (K[2])+1)dK[2]+c_1\right ) \]