Internal problem ID [2931]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 7
Problem number: 183.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{m}+b y+c \,x^{n} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 174
dsolve(x*diff(y(x),x) = a*x^m-b*y(x)-c*x^n*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\BesselY \left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1}+\BesselJ \left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) x^{\frac {n}{2}+\frac {m}{2}} \sqrt {-a c}\, x^{-n +1}}{\left (\BesselY \left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1}+\BesselJ \left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) c x} \]
✓ Solution by Mathematica
Time used: 0.855 (sec). Leaf size: 433
DSolve[x y'[x]==a x^m-b y[x]-c x^n y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(m+n) x^{-n} \left ((-1)^{\frac {n}{m+n}} (m+n)^{\frac {2 n}{m+n}} \left ((m+n)^2\right )^{\frac {b}{m+n}} \text {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \, _0\tilde {F}_1\left (;\frac {n-b}{m+n};\frac {a c x^{m+n}}{(m+n)^2}\right )+\frac {c_1 \left ((m+n)^2\right )^{\frac {n}{m+n}} (-1)^{\frac {b}{m+n}} (m+n)^{\frac {2 b}{m+n}+1} \left (\frac {\sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )^{\frac {2 (b+m)}{m+n}} \, _0F_1\left (;\frac {b+m}{m+n}+1;\frac {a c x^{m+n}}{(m+n)^2}\right )}{b+m}\right )}{c \left ((-1)^{\frac {n}{m+n}} (m+n)^{\frac {2 n}{m+n}} \left ((m+n)^2\right )^{\frac {b}{m+n}} \, _0F_1\left (;\frac {n-b}{m+n}+1;\frac {a c x^{m+n}}{(m+n)^2}\right )+c_1 \left ((m+n)^2\right )^{\frac {n}{m+n}} (-1)^{\frac {b}{m+n}} (m+n)^{\frac {2 b}{m+n}} \left (\frac {\sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )^{\frac {2 (b-n)}{m+n}} \, _0F_1\left (;\frac {b+m}{m+n};\frac {a c x^{m+n}}{(m+n)^2}\right )\right )} \\ y(x)\to \frac {a x^m \, _0\tilde {F}_1\left (;\frac {b+m}{m+n}+1;\frac {a c x^{m+n}}{(m+n)^2}\right )}{(m+n) \, _0\tilde {F}_1\left (;\frac {b+m}{m+n};\frac {a c x^{m+n}}{(m+n)^2}\right )} \\ \end{align*}