Internal problem ID [13349]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 5. LINEAR FIRST ORDER EQUATIONS. Additional exercises. page
103
Problem number: 5.1 (c).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} x=\sqrt {x}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
dsolve(diff(y(x),x)-x*y(x)^2=sqrt(x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )}{x^{\frac {1}{4}} \left (\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{\frac {7}{4}}}{7}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.242 (sec). Leaf size: 273
DSolve[y'[x]-x*y[x]^2==Sqrt[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^{7/4} \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (-\frac {3}{7},\frac {4 x^{7/4}}{7}\right )-x^{7/4} \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {11}{7},\frac {4 x^{7/4}}{7}\right )+2 \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {4}{7},\frac {4 x^{7/4}}{7}\right )+c_1 x^{7/4} \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {11}{7},\frac {4 x^{7/4}}{7}\right )-c_1 x^{7/4} \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (\frac {3}{7},\frac {4 x^{7/4}}{7}\right )+2 c_1 \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )}{2 x^2 \left (\operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {4}{7},\frac {4 x^{7/4}}{7}\right )+c_1 \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )\right )} \\ y(x)\to -\frac {x^{7/4} \operatorname {BesselJ}\left (-\frac {11}{7},\frac {4 x^{7/4}}{7}\right )-x^{7/4} \operatorname {BesselJ}\left (\frac {3}{7},\frac {4 x^{7/4}}{7}\right )+2 \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )}{2 x^2 \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )} \\ \end{align*}