61.2.60 problem 60

Internal problem ID [12066]
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 : 60
Date solved : Tuesday, January 28, 2025 at 06:50:23 PM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (a \,x^{2}+b x +c \right ) y^{\prime }&=y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 6204

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

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 7.439 (sec). Leaf size: 245

DSolve[(a*x^2+b*x+c)*D[y[x],x]==y[x]^2+(a*x+\[Mu])*y[x]-\[Lambda]^2*x^2+\[Lambda]*(b-\[Mu])*x+\[Lambda]*c,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {c \left (-\exp \left (\int _1^x\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )\right )+\lambda x \int _1^x\exp \left (\int _1^{K[2]}\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )dK[2]-x \left (b \exp \left (\int _1^x\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )+a x \exp \left (\int _1^x\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )-c_1 \lambda \right )}{\int _1^x\exp \left (\int _1^{K[2]}\frac {-b+\mu -a K[1]+2 \lambda K[1]}{c+K[1] (b+a K[1])}dK[1]\right )dK[2]+c_1} \\ y(x)\to \lambda x \\ \end{align*}