7.2.18 problem 20
Internal
problem
ID
[36]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.3.
Problems
at
page
27
Problem
number
:
20
Date
solved
:
Friday, February 07, 2025 at 08:12:34 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Solution by Maple
Time used: 0.270 (sec). Leaf size: 157
dsolve([diff(y(x),x)=x^2-y(x)^2,y(0) = 1],y(x), singsol=all)
\[
y = \left \{\begin {array}{cc} \frac {\left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \pi \left (-\Gamma \left (\frac {3}{4}\right )^{2} \sqrt {2}+\pi \right )+2 \operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right ) x}{\pi \left (-\Gamma \left (\frac {3}{4}\right )^{2} \sqrt {2}+\pi \right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )-2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & x <0 \\ 1 & x =0 \\ \frac {\left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \pi \left (\Gamma \left (\frac {3}{4}\right )^{2} \sqrt {2}+\pi \right )-2 \operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right ) x}{\pi \left (\Gamma \left (\frac {3}{4}\right )^{2} \sqrt {2}+\pi \right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & 0<x \end {array}\right .
\]
✓ Solution by Mathematica
Time used: 0.595 (sec). Leaf size: 151
DSolve[{D[y[x],x]==x^2-y[x]^2,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {2 i x^2 \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )+(1+i) \sqrt {2} \operatorname {Gamma}\left (\frac {3}{4}\right ) \left (i x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )-i x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right )}{2 x \left ((1+i) \sqrt {2} \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )+\operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )\right )}
\]