Internal problem ID [3640]
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]
\[ \boxed {y^{\prime } x^{n}-b^{2} y^{2}=a^{2} x^{-2+2 n}} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 67
dsolve(x^n*diff(y(x),x) = a^2*x^(2*n-2)+b^2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{n -1} \left (n -1+\tan \left (\frac {\sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a^{2} b^{2}-n^{2}+2 n -1}\right )}{2 b^{2}} \]
✓ Solution by Mathematica
Time used: 0.541 (sec). Leaf size: 162
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 (\left (-a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}+n-1\right ) x^{a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}}+c_1 \left (a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}+n-1\right )\right )}{2 b^2 \left (x^{a b \sqrt {\frac {(n-1)^2}{a^2 b^2}-4}}+c_1\right )} \\ 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*}