61.2.2 problem 2

Internal problem ID [12008]
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
Date solved : Monday, January 27, 2025 at 11:48:12 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-a^{2} x^{2}+3 a \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 82

dsolve(diff(y(x),x)=y(x)^2-a^2*x^2+3*a,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{a \,x^{2}} c_{1} a x -c_{1} \sqrt {\pi }\, \left (x^{2} \left (-a \right )^{{3}/{2}}+\sqrt {-a}\right ) \operatorname {erf}\left (\sqrt {-a}\, x \right )+a \,x^{2}-1}{\sqrt {\pi }\, \sqrt {-a}\, \operatorname {erf}\left (\sqrt {-a}\, x \right ) c_{1} x +{\mathrm e}^{a \,x^{2}} c_{1} +x} \]

✓ Solution by Mathematica

Time used: 0.481 (sec). Leaf size: 192

DSolve[D[y[x],x]==y[x]^2-a^2*x^2+3*a,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {a x \operatorname {ParabolicCylinderD}\left (-2,i \sqrt {2} \sqrt {a} x\right )+i \sqrt {2} \sqrt {a} \operatorname {ParabolicCylinderD}\left (-1,i \sqrt {2} \sqrt {a} x\right )-a c_1 x \operatorname {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right )+\sqrt {2} \sqrt {a} c_1 \operatorname {ParabolicCylinderD}\left (2,\sqrt {2} \sqrt {a} x\right )}{\operatorname {ParabolicCylinderD}\left (-2,i \sqrt {2} \sqrt {a} x\right )+c_1 \operatorname {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right )} \\ y(x)\to \frac {\sqrt {2} \sqrt {a} \operatorname {ParabolicCylinderD}\left (2,\sqrt {2} \sqrt {a} x\right )}{\operatorname {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right )}-a x \\ \end{align*}