Internal problem ID [10358]
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]
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y=-a \,x^{n} b -b^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 74
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)-a*b*x^n-b^2,y(x), singsol=all)
\[ \frac {\left (b -y \left (x \right )\right ) \left (\int _{}^{x}{\mathrm e}^{\frac {\left (\textit {\_a}^{n} a +2 b \left (n +1\right )\right ) \textit {\_a}}{n +1}}d \textit {\_a} \right )+c_{1} b -c_{1} y \left (x \right )-{\mathrm e}^{\frac {\left (a \,x^{n}+2 b \left (n +1\right )\right ) x}{n +1}}}{b -y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.948 (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]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {a x^{n+1}}{n+1}+2 b x}}{a n (K[2]-b)^2}-\int _1^x\left (\frac {e^{\frac {a K[1]^{n+1}}{n+1}+2 b K[1]} \left (a K[1]^n+b+K[2]\right )}{a n (b-K[2])^2}+\frac {e^{\frac {a K[1]^{n+1}}{n+1}+2 b K[1]}}{a n (b-K[2])}\right )dK[1]\right )dK[2]+\int _1^x\frac {e^{\frac {a K[1]^{n+1}}{n+1}+2 b K[1]} \left (a K[1]^n+b+y(x)\right )}{a n (b-y(x))}dK[1]=c_1,y(x)\right ] \]