Internal problem ID [3133]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 386.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x^{n}-x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 42
dsolve(x^n*diff(y(x),x) = x^(n-1)*(a*x^(2*n)+n*y(x)-b*y(x)^2),y(x), singsol=all)
\[ y \relax (x ) = -\frac {i \tan \left (\frac {i x^{n} \sqrt {b}\, \sqrt {a}-c_{1} n}{n}\right ) \sqrt {a}\, x^{n}}{\sqrt {b}} \]
✓ Solution by Mathematica
Time used: 0.319 (sec). Leaf size: 145
DSolve[x^n y'[x]==x^(n-1)(a x^(2 n)+n y[x]-b y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} x^n \left (-\cosh \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )+c_1 \sin \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )\right )}{\sqrt {-b} \left (\sin \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )+c_1 \cosh \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )\right )} \\ y(x)\to \frac {\sqrt {a} x^n \tanh \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )}{\sqrt {b}} \\ \end{align*}