Internal problem ID [7595]
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]]
Solve \begin {gather*} \boxed {y^{\prime }+y^{2}+a \,x^{m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 189
dsolve(diff(y(x),x) + y(x)^2 + a*x^m=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\BesselJ \left (\frac {m +3}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right ) \sqrt {a}\, x^{\frac {m}{2}+1} c_{1}+\BesselY \left (\frac {m +3}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right ) \sqrt {a}\, x^{\frac {m}{2}+1}-c_{1} \BesselJ \left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )-\BesselY \left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )}{x \left (c_{1} \BesselJ \left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )+\BesselY \left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.239 (sec). Leaf size: 319
DSolve[y'[x] + y[x]^2 + a*x^m==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(m+2) \left (\frac {i \sqrt {-a} x^{\frac {m}{2}+1}}{m+2}\right )^{\frac {2}{m+2}} \, _0\tilde {F}_1\left (;\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )+c_1 (m+2) \, _0\tilde {F}_1\left (;-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )+c_1 \, _0\tilde {F}_1\left (;\frac {m+1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}{x \left (\left (\frac {i \sqrt {-a} x^{\frac {m}{2}+1}}{m+2}\right )^{\frac {2}{m+2}} \, _0\tilde {F}_1\left (;1+\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )+c_1 \, _0\tilde {F}_1\left (;\frac {m+1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )\right )} \\ y(x)\to \frac {1-\frac {\, _0F_1\left (;-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}{\, _0F_1\left (;\frac {m+1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}}{x} \\ y(x)\to \frac {1-\frac {\, _0F_1\left (;-\frac {1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}{\, _0F_1\left (;\frac {m+1}{m+2};-\frac {a x^{m+2}}{(m+2)^2}\right )}}{x} \\ \end{align*}