29.2.14 problem 39
Internal
problem
ID
[4647]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
2
Problem
number
:
39
Date
solved
:
Monday, January 27, 2025 at 09:29:20 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 44
dsolve(diff(y(x),x) = x^2-y(x)^2,y(x), singsol=all)
\[
y \left (x \right ) = \frac {x \left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_{1} -\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{c_{1} \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )}
\]
✓ Solution by Mathematica
Time used: 0.168 (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*}