29.14.6 problem 386

Internal problem ID [4985]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 386
Date solved : Monday, January 27, 2025 at 10:01:15 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x^{n} y^{\prime }&=x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 34

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)
 

\[ y \left (x \right ) = \frac {\tanh \left (\frac {x^{n} \sqrt {a}\, \sqrt {b}+i c_{1} n}{n}\right ) \sqrt {a}\, x^{n}}{\sqrt {b}} \]

✓ Solution by Mathematica

Time used: 0.320 (sec). Leaf size: 153

DSolve[x^n D[y[x],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 (-\cos \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 \cos \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )\right )} \\ y(x)\to \frac {\sqrt {a} x^n \tan \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )}{\sqrt {-b}} \\ \end{align*}