Internal problem ID [9589]
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: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+a^{2} x^{2}-3 a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 108
dsolve(diff(y(x),x)=y(x)^2-a^2*x^2+3*a,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\frac {a \,x^{2}}{2}} c_{1} a x +\left (\left (-\sqrt {\pi }\, \left (-a \right )^{\frac {3}{2}} c_{1} x^{2}-\sqrt {\pi }\, \sqrt {-a}\, c_{1}\right ) \erf \left (\sqrt {-a}\, x \right )+a \,x^{2}-1\right ) {\mathrm e}^{-\frac {a \,x^{2}}{2}}}{{\mathrm e}^{\frac {a \,x^{2}}{2}} c_{1}+\left (\erf \left (\sqrt {-a}\, x \right ) \sqrt {\pi }\, \sqrt {-a}\, c_{1} x +x \right ) {\mathrm e}^{-\frac {a \,x^{2}}{2}}} \]
✓ Solution by Mathematica
Time used: 0.336 (sec). Leaf size: 94
DSolve[y'[x]==y[x]^2-a^2*x^2+3*a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a x+\frac {\sqrt {a} \left (\sqrt {\pi } \left (\text {Erfi}\left (\sqrt {a} x\right )+i\right )-\sqrt {2} c_1\right )}{2 e^{a x^2} H_{-2}\left (i \sqrt {a} x\right )+\sqrt {2} \sqrt {a} c_1 x} \\ y(x)\to a x-\frac {1}{x} \\ \end{align*}