Internal problem ID [3130]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 382.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x^{n}-x^{2 n -1}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 52
dsolve(x^n*diff(y(x),x) = x^(2*n-1)-y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (-\BesselK \left (n , 2 \sqrt {x}\right ) c_{1}+\BesselI \left (n , 2 \sqrt {x}\right )\right ) x^{n}}{\sqrt {x}\, \left (\BesselK \left (n -1, 2 \sqrt {x}\right ) c_{1}+\BesselI \left (n -1, 2 \sqrt {x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.333 (sec). Leaf size: 80
DSolve[x^n y'[x]==x^(2 n -1)-y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^n \left ((n-1) \, _0F_1(;1-n;x)+c_1 (-1)^n x^n \text {Gamma}(n) \, _0\tilde {F}_1(;n+1;x)\right )}{-x \, _0F_1(;2-n;x)+c_1 (-1)^n x^n \, _0F_1(;n;x)} \\ y(x)\to \frac {x^n \, _0\tilde {F}_1(;n+1;x)}{\, _0\tilde {F}_1(;n;x)} \\ \end{align*}