61.2.61 problem 61

Internal problem ID [12067]
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 : 61
Date solved : Tuesday, January 28, 2025 at 12:11:48 AM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }&=y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2} \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 5768

dsolve((a__2*x^2+b__2*x+c__2)*diff(y(x),x)=y(x)^2+(a__1*x+b__1)*y(x)-lambda*(lambda+a__1-a__2)*x^2+lambda*(b__2-b__1)*x+lambda*c__2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 9.150 (sec). Leaf size: 250

DSolve[(a2*x^2+b2*x+c2)*D[y[x],x]==y[x]^2+(a1*x+b1)*y[x]-\[Lambda]*(\[Lambda]+a1-a2)*x^2+\[Lambda]*(b2-b1)*x+\[Lambda]*c2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\text {c2} \left (-\exp \left (\int _1^x\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )\right )+\lambda x \int _1^x\exp \left (\int _1^{K[2]}\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )dK[2]-x \left (\text {b2} \exp \left (\int _1^x\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )+\text {a2} x \exp \left (\int _1^x\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )-c_1 \lambda \right )}{\int _1^x\exp \left (\int _1^{K[2]}\frac {\text {b1}-\text {b2}+(\text {a1}-2 \text {a2}+2 \lambda ) K[1]}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]\right )dK[2]+c_1} \\ y(x)\to \lambda x \\ \end{align*}