Internal problem ID [2813]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 58.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{2}-b y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 71
dsolve(diff(y(x),x) = a*x^2+b*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\sqrt {a b}\, x \left (\BesselJ \left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1}+\BesselY \left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )}{b \left (c_{1} \BesselJ \left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\BesselY \left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.14 (sec). Leaf size: 153
DSolve[y'[x]==a x^2+b y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} x \left (-J_{-\frac {3}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 J_{\frac {3}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )}{\sqrt {b} \left (J_{\frac {1}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 J_{-\frac {1}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )} \\ y(x)\to \frac {a x^3 \, _0\tilde {F}_1\left (;\frac {7}{4};-\frac {1}{16} a b x^4\right )}{4 \, _0\tilde {F}_1\left (;\frac {3}{4};-\frac {1}{16} a b x^4\right )} \\ \end{align*}