29.6.22 problem 168

Internal problem ID [4768]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 6
Problem number : 168
Date solved : Monday, January 27, 2025 at 09:37:02 AM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.316 (sec). Leaf size: 139

DSolve[x D[y[x],x]==a x^(2 n)+(n+b y[x])y[x],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*}