44.3.12 problem 20
Internal
problem
ID
[6993]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
1.
Introduction
to
differential
equations.
Review
problems
at
page
34
Problem
number
:
20
Date
solved
:
Monday, January 27, 2025 at 02:40:10 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=x^{2}+y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (1\right )&=-1 \end{align*}
✓ Solution by Maple
Time used: 0.118 (sec). Leaf size: 79
dsolve([diff(y(x),x)=x^2+y(x)^2,y(1) = -1],y(x), singsol=all)
\[
y = -\frac {x \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {1}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {1}{2}\right )\right )+\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {1}{2}\right )-\operatorname {BesselJ}\left (\frac {1}{4}, \frac {1}{2}\right )\right )\right )}{\left (-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {1}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {1}{2}\right )\right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {1}{2}\right )-\operatorname {BesselJ}\left (\frac {1}{4}, \frac {1}{2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 199
DSolve[{D[y[x],x]==x^2+y[x]^2,{y[1]==-1}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {x^2 \left (\operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2}\right )-\operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2}\right )-\operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2}\right )\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {x^2}{2}\right )+x^2 \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2}\right )-\operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2}\right )\right ) \operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )-\left (\operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2}\right )-\operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2}\right )\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )-x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )\right )}{x \left (2 \left (\operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2}\right )-\operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2}\right )\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )+\left (-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2}\right )+\operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2}\right )\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {x^2}{2}\right )\right )}
\]