29.13.14 problem 368

Internal problem ID [4968]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 13
Problem number : 368
Date solved : Monday, January 27, 2025 at 10:00:12 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

\begin{align*} x \left (c \,x^{2}+b x +a \right ) y^{\prime }+x^{2}-\left (c \,x^{2}+b x +a \right ) y&=y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 58

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

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

✓ Solution by Mathematica

Time used: 1.247 (sec). Leaf size: 116

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

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