Internal problem ID [7594]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {y^{\prime }+y^{2}+a \,x^{m}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 187
dsolve(diff(y(x),x) + y(x)^2 + a*x^m=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\sqrt {a}\, x^{\frac {m}{2}+1} \operatorname {BesselJ}\left (\frac {m +3}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right ) c_{1} -\operatorname {BesselY}\left (\frac {m +3}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right ) \sqrt {a}\, x^{\frac {m}{2}+1}+c_{1} \operatorname {BesselJ}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )+\operatorname {BesselY}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )}{x \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )+\operatorname {BesselY}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.359 (sec). Leaf size: 470
DSolve[y'[x] + y[x]^2 + a*x^m==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} x^{\frac {m}{2}+1} \left (2 \operatorname {Gamma}\left (1+\frac {1}{m+2}\right ) \operatorname {BesselJ}\left (\frac {1}{m+2}-1,\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )+c_1 \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \left (\operatorname {BesselJ}\left (-\frac {m+3}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )-\operatorname {BesselJ}\left (\frac {m+1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )\right )\right )+c_1 \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )}{2 x \left (\operatorname {Gamma}\left (1+\frac {1}{m+2}\right ) \operatorname {BesselJ}\left (\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )+c_1 \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )\right )} \\ y(x)\to \frac {\frac {(m+2) \left (\, _0\tilde {F}_1\left (;-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )-\frac {a x^{m+2} \, _0\tilde {F}_1\left (;2-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}{(m+2)^2}\right )}{\, _0\tilde {F}_1\left (;\frac {m+1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}+1}{2 x} \\ y(x)\to \frac {\frac {(m+2) \left (\, _0\tilde {F}_1\left (;-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )-\frac {a x^{m+2} \, _0\tilde {F}_1\left (;2-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}{(m+2)^2}\right )}{\, _0\tilde {F}_1\left (;\frac {m+1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}+1}{2 x} \\ \end{align*}