Internal problem ID [3439]
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]
\[ \boxed {x y^{\prime }+y b +c \,x^{n} y^{2}=a \,x^{m}} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = -\frac {\left (\operatorname {BesselY}\left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) x^{\frac {n}{2}+\frac {m}{2}} \sqrt {-a c}\, x^{-n +1}}{\left (\operatorname {BesselY}\left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right ) c x} \]
✓ Solution by Mathematica
Time used: 0.973 (sec). Leaf size: 1549
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 {x^{-n} \left ((-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (-\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (-\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {n-b}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {n}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 n}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {n-b}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} (m+n)^{\frac {2 b}{m+n}+1} x^{m+n} \operatorname {BesselI}\left (\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} m (m+n)^{\frac {2 b}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {b-n}{m+n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}+(-1)^{\frac {b}{m+n}} \sqrt {a} \sqrt {c} n (m+n)^{\frac {2 b}{m+n}} x^{m+n} \operatorname {BesselI}\left (\frac {b-n}{m+n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}-(-1)^{\frac {n}{m+n}} b (m+n)^{\frac {2 n}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {n-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}+\frac {1}{2}}+(-1)^{\frac {n}{m+n}} n (m+n)^{\frac {2 n}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {n-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}+\frac {1}{2}}+(-1)^{\frac {b}{m+n}} (n-b) (m+n)^{\frac {2 b}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}+\frac {1}{2}}\right )}{2 c \sqrt {(m+n)^2} \sqrt {x^{m+n}} \left ((-1)^{\frac {n}{m+n}} (m+n)^{\frac {2 n}{m+n}} \operatorname {BesselI}\left (\frac {n-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {-b+m+2 n}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b}{m+n}}+(-1)^{\frac {b}{m+n}} (m+n)^{\frac {2 b}{m+n}} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {b+m}{m+n}\right ) \left ((m+n)^2\right )^{\frac {n}{m+n}}\right )} y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {b+m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )+(n-b) \sqrt {(m+n)^2} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )+\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselI}\left (\frac {b-n}{m+n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )\right )}{2 c \sqrt {(m+n)^2} \operatorname {BesselI}\left (\frac {b-n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )} \end{align*}