29.10.2 problem 268

Internal problem ID [4868]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 10
Problem number : 268
Date solved : Monday, January 27, 2025 at 09:45:40 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 59

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

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

✓ Solution by Mathematica

Time used: 0.267 (sec). Leaf size: 99

DSolve[x^2 D[y[x],x]==a+b x y[x]+c x^2 y[x]^2,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 c x} \\ y(x)\to -\frac {-\sqrt {-4 a c+b^2+2 b+1}+b+1}{2 c x} \\ \end{align*}