Internal problem ID [3132]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 385.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x^{n}-a^{2} x^{2 n -2}-b^{2} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 88
dsolve(x^n*diff(y(x),x) = a^2*x^(2*n-2)+b^2*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (-\tan \left (-\frac {\ln \relax (x ) \sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}}{2}+\frac {c_{1} \sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}}{2}\right ) \sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}+n -1\right ) x^{n -1}}{2 b^{2}} \]
✓ Solution by Mathematica
Time used: 0.465 (sec). Leaf size: 115
DSolve[x^n y'[x]==a^2 x^(2 n-2)+b^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{n-1} \left (a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4} \left (-1+\frac {2 c_1}{x^{a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}}+c_1}\right )+n-1\right )}{2 b^2} \\ y(x)\to \frac {x^{n-1} \left (a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}+n-1\right )}{2 b^2} \\ \end{align*}