8.2.16 problem 20

Internal problem ID [676]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.3. Slope fields and solution curves. Page 26
Problem number : 20
Date solved : Monday, January 27, 2025 at 02:57:29 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 44

dsolve(diff(y(x),x) = x^2-y(x)^2,y(x), singsol=all)
 

\[ y = \frac {x \left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_1 -\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{c_1 \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.119 (sec). Leaf size: 197

DSolve[D[y[x],x]== x^2-y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {-i x^2 \left (2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )+c_1 \left (\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )-\operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right )} \\ y(x)\to \frac {i x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )-i x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )} \\ \end{align*}