Internal problem ID [9615]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 28.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \,x^{n} y+a \,x^{n} b +b^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 89
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)-a*b*x^n-b^2,y(x), singsol=all)
\[ c_{1}+\int _{}^{x}-\frac {\left (-b \textit {\_a} +\textit {\_a} y \relax (x )\right ) {\mathrm e}^{\frac {2 b \textit {\_a} n +\textit {\_a}^{n +1} a +2 b \textit {\_a}}{n +1}}}{\left (b -y \relax (x )\right ) \textit {\_a}}d \textit {\_a} -\frac {{\mathrm e}^{\frac {2 b x n +x^{n +1} a +2 b x}{n +1}}}{b -y \relax (x )} = 0 \]
✓ Solution by Mathematica
Time used: 1.863 (sec). Leaf size: 195
DSolve[y'[x]==y[x]^2+a*x^n*y[x]-a*b*x^n-b^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {a x^{n+1}}{n+1}+2 b x}}{a n (K[2]-b)^2}-\int _1^x\left (\frac {e^{\frac {a K[1]^{n+1}}{n+1}+2 b K[1]} \left (a K[1]^n+b+K[2]\right )}{a n (b-K[2])^2}+\frac {e^{\frac {a K[1]^{n+1}}{n+1}+2 b K[1]}}{a n (b-K[2])}\right )dK[1]\right )dK[2]+\int _1^x\frac {e^{\frac {a K[1]^{n+1}}{n+1}+2 b K[1]} \left (a K[1]^n+b+y(x)\right )}{a n (b-y(x))}dK[1]=c_1,y(x)\right ] \]