29.7.8 problem 183

Internal problem ID [4783]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 183
Date solved : Monday, January 27, 2025 at 09:37:41 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }&=a \,x^{m}-b y-c \,x^{n} y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 166

dsolve(x*diff(y(x),x) = a*x^m-b*y(x)-c*x^n*y(x)^2,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {-a c}\, \left (-\operatorname {BesselY}\left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {b +m}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right )}{c \left (\operatorname {BesselY}\left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b -n}{n +m}, \frac {2 \sqrt {-a c}\, x^{\frac {n}{2}+\frac {m}{2}}}{n +m}\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.802 (sec). Leaf size: 1549

DSolve[x D[y[x],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*}