61.2.13 problem 13

Internal problem ID [12019]
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 : 13
Date solved : Monday, January 27, 2025 at 11:50:59 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Riccati, _special]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 40

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

\[ y = \frac {-1+\tan \left (\frac {\sqrt {4 a b -1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a b -1}}{2 a x} \]

✓ Solution by Mathematica

Time used: 0.190 (sec). Leaf size: 77

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

\begin{align*} y(x)\to \frac {-1+\sqrt {1-4 a b} \left (-1+\frac {2 c_1}{x^{\sqrt {1-4 a b}}+c_1}\right )}{2 a x} \\ y(x)\to \frac {\sqrt {1-4 a b}-1}{2 a x} \\ \end{align*}