Internal problem ID [10401]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }-y^{2}-\left (a_{1} x +b_{1} \right ) y=-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 545907
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: 34.34 (sec). Leaf size: 284
DSolve[(a2*x^2+b2*x+c2)*y'[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 {\lambda x \int _1^x\exp \left (\frac {(\text {a1}-2 \text {a2}+2 \lambda ) \log (\text {c2}+K[1] (\text {b2}+\text {a2} K[1]))-\frac {2 (\text {b2} (\text {a1}+2 \lambda )-2 \text {a2} \text {b1}) \arctan \left (\frac {\text {b2}+2 \text {a2} K[1]}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}\right )}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}}{2 \text {a2}}\right )dK[1]+(x (\text {a2} x+\text {b2})+\text {c2})^{\frac {\text {a1}+2 \lambda }{2 \text {a2}}} \left (-\exp \left (\frac {(2 \text {a2} \text {b1}-\text {b2} (\text {a1}+2 \lambda )) \arctan \left (\frac {2 \text {a2} x+\text {b2}}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}\right )}{\text {a2} \sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}\right )\right )+c_1 \lambda x}{\int _1^x\exp \left (\frac {(\text {a1}-2 \text {a2}+2 \lambda ) \log (\text {c2}+K[1] (\text {b2}+\text {a2} K[1]))-\frac {2 (\text {b2} (\text {a1}+2 \lambda )-2 \text {a2} \text {b1}) \arctan \left (\frac {\text {b2}+2 \text {a2} K[1]}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}\right )}{\sqrt {4 \text {a2} \text {c2}-\text {b2}^2}}}{2 \text {a2}}\right )dK[1]+c_1} y(x)\to \lambda x \end{align*}