Internal problem ID [3638]
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]
\[ \boxed {y^{\prime } x^{n}+y^{2}=x^{2 n -1}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 80
dsolve(x^n*diff(y(x),x) = x^(2*n-1)-y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\operatorname {BesselK}\left (n , 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselI}\left (n , 2 \sqrt {x}\right )\right ) x^{n}}{-\operatorname {BesselI}\left (n +1, 2 \sqrt {x}\right ) \sqrt {x}-\sqrt {x}\, \operatorname {BesselK}\left (n +1, 2 \sqrt {x}\right ) c_{1} +n \left (\operatorname {BesselK}\left (n , 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselI}\left (n , 2 \sqrt {x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.366 (sec). Leaf size: 293
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-1} \left (-\left ((n-1) \operatorname {Gamma}(2-n) \operatorname {BesselI}\left (1-n,2 \sqrt {x}\right )\right )+\sqrt {x} \operatorname {Gamma}(2-n) \operatorname {BesselI}\left (2-n,2 \sqrt {x}\right )+\sqrt {x} \operatorname {Gamma}(2-n) \operatorname {BesselI}\left (-n,2 \sqrt {x}\right )-c_1 (-1)^n \sqrt {x} \operatorname {Gamma}(n) \operatorname {BesselI}\left (n-2,2 \sqrt {x}\right )-c_1 (-1)^n \operatorname {Gamma}(n) \operatorname {BesselI}\left (n-1,2 \sqrt {x}\right )+c_1 (-1)^n n \operatorname {Gamma}(n) \operatorname {BesselI}\left (n-1,2 \sqrt {x}\right )-c_1 (-1)^n \sqrt {x} \operatorname {Gamma}(n) \operatorname {BesselI}\left (n,2 \sqrt {x}\right )\right )}{2 \left (\operatorname {Gamma}(2-n) \operatorname {BesselI}\left (1-n,2 \sqrt {x}\right )-c_1 (-1)^n \operatorname {Gamma}(n) \operatorname {BesselI}\left (n-1,2 \sqrt {x}\right )\right )} \\ y(x)\to \frac {x^{n-1} \left (\sqrt {x} \operatorname {BesselI}\left (n-2,2 \sqrt {x}\right )-(n-1) \operatorname {BesselI}\left (n-1,2 \sqrt {x}\right )+\sqrt {x} \operatorname {BesselI}\left (n,2 \sqrt {x}\right )\right )}{2 \operatorname {BesselI}\left (n-1,2 \sqrt {x}\right )} \\ \end{align*}