Internal problem ID [9742]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \ln \left (\beta x \right ) y+a b \ln \left (\beta x \right )+b^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 55
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)
\[ y \relax (x ) = b -\frac {\left (\beta x \right )^{x a} {\mathrm e}^{-x a} {\mathrm e}^{2 b x}}{\int \left (\beta x \right )^{x a} {\mathrm e}^{-x a} {\mathrm e}^{2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 0.907 (sec). Leaf size: 187
DSolve[y'[x]==y[x]^2+a*Log[\[Beta]*x]*y[x]-a*b*Log[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {e^{2 b K[1]-a K[1]} (\beta K[1])^{a K[1]} (b+a \log (\beta K[1])+y(x))}{a (b-y(x))}dK[1]+\int _1^{y(x)}\left (\frac {e^{2 b x-a x} (x \beta )^{a x}}{a (K[2]-b)^2}-\int _1^x\left (\frac {e^{2 b K[1]-a K[1]} (b+K[2]+a \log (\beta K[1])) (\beta K[1])^{a K[1]}}{a (b-K[2])^2}+\frac {e^{2 b K[1]-a K[1]} (\beta K[1])^{a K[1]}}{a (b-K[2])}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]