61.2.49 problem 49

Internal problem ID [12055]
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 : 49
Date solved : Monday, January 27, 2025 at 11:54:59 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 59

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

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

✓ Solution by Mathematica

Time used: 0.260 (sec). Leaf size: 99

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

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