61.2.30 problem 30

Internal problem ID [12036]
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 : 30
Date solved : Tuesday, January 28, 2025 at 06:41:20 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 98

dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*x^m*y(x)+b*c*x^m-a*c^2*x^n,y(x), singsol=all)
 

\[ \frac {a \left (c +y\right ) \left (\int _{}^{x}{\mathrm e}^{-\frac {2 \textit {\_a} \left (-\frac {b \left (n +1\right ) \textit {\_a}^{m}}{2}+a \,\textit {\_a}^{n} c \left (m +1\right )\right )}{\left (m +1\right ) \left (n +1\right )}} \textit {\_a}^{n}d \textit {\_a} \right )+c_{1} y+c_{1} c +{\mathrm e}^{-\frac {2 x \left (-\frac {b \left (n +1\right ) x^{m}}{2}+a \,x^{n} c \left (m +1\right )\right )}{\left (m +1\right ) \left (n +1\right )}}}{c +y} = 0 \]

✓ Solution by Mathematica

Time used: 1.548 (sec). Leaf size: 286

DSolve[D[y[x],x]==a*x^n*y[x]^2+b*x^m*y[x]+b*c*x^m-a*c^2*x^n,y[x],x,IncludeSingularSolutions -> True]
 

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