61.8.1 problem 10

Internal problem ID [12161]
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 : 10
Date solved : Tuesday, January 28, 2025 at 12:45:54 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 73

dsolve(diff(y(x),x)=y(x)^2+a*ln(beta*x)*y(x)-a*b*ln(beta*x)-b^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.602 (sec). Leaf size: 187

DSolve[D[y[x],x]==y[x]^2+a*Log[\[Beta]*x]*y[x]-a*b*Log[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {e^{2 b K[1]-a K[1]} (\beta K[1])^{a K[1]} (b+a \log (\beta K[1])+y(x))}{a (b-y(x))}dK[1]+\int _1^{y(x)}\left (\frac {e^{2 b x-a x} (x \beta )^{a x}}{a (K[2]-b)^2}-\int _1^x\left (\frac {e^{2 b K[1]-a K[1]} (b+K[2]+a \log (\beta K[1])) (\beta K[1])^{a K[1]}}{a (b-K[2])^2}+\frac {e^{2 b K[1]-a K[1]} (\beta K[1])^{a K[1]}}{a (b-K[2])}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]