29.21.12 problem 588

Internal problem ID [5182]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 588
Date solved : Monday, January 27, 2025 at 10:18:15 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 82

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

\begin{align*} y \left (x \right ) &= \frac {\sqrt {-B \left (-x^{\frac {2 B}{a}} \left (b \,x^{2}+a \right )^{-\frac {B}{a}} c_{1} B +A \right )}}{B} \\ y \left (x \right ) &= -\frac {\sqrt {-B \left (-x^{\frac {2 B}{a}} \left (b \,x^{2}+a \right )^{-\frac {B}{a}} c_{1} B +A \right )}}{B} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.966 (sec). Leaf size: 134

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

\begin{align*} y(x)\to -\frac {\sqrt {-A+e^{2 B c_1} x^{\frac {2 B}{a}} \left (a+b x^2\right )^{-\frac {B}{a}}}}{\sqrt {B}} \\ y(x)\to \frac {\sqrt {-A+e^{2 B c_1} x^{\frac {2 B}{a}} \left (a+b x^2\right )^{-\frac {B}{a}}}}{\sqrt {B}} \\ y(x)\to -\frac {i \sqrt {A}}{\sqrt {B}} \\ y(x)\to \frac {i \sqrt {A}}{\sqrt {B}} \\ \end{align*}