44.4.1 problem 1 (a)
Internal
problem
ID
[7014]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.1
Solution
curves
without
a
solution.
Exercises
2.1
at
page
44
Problem
number
:
1
(a)
Date
solved
:
Sunday, March 30, 2025 at 11:34:01 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (-2\right )&=1 \end{align*}
✓ Maple. Time used: 0.129 (sec). Leaf size: 84
ode:=diff(y(x),x) = x^2-y(x)^2;
ic:=y(-2) = 1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = -\frac {2 x \left (\left (\frac {\operatorname {BesselK}\left (\frac {1}{4}, 2\right )}{2}-\operatorname {BesselK}\left (\frac {3}{4}, 2\right )\right ) \operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (\operatorname {BesselI}\left (-\frac {3}{4}, 2\right )+\frac {\operatorname {BesselI}\left (\frac {1}{4}, 2\right )}{2}\right )\right )}{\left (2 \operatorname {BesselK}\left (\frac {3}{4}, 2\right )-\operatorname {BesselK}\left (\frac {1}{4}, 2\right )\right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+2 \left (\operatorname {BesselI}\left (-\frac {3}{4}, 2\right )+\frac {\operatorname {BesselI}\left (\frac {1}{4}, 2\right )}{2}\right ) \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )}
\]
✓ Mathematica. Time used: 0.115 (sec). Leaf size: 235
ode=D[y[x],x]==x^2-y[x]^2;
ic={y[-2]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {x^2 \left (-4 i \operatorname {BesselJ}\left (-\frac {5}{4},2 i\right )-5 \operatorname {BesselJ}\left (-\frac {1}{4},2 i\right )+4 i \operatorname {BesselJ}\left (\frac {3}{4},2 i\right )\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )+2 x^2 \left (2 i \operatorname {BesselJ}\left (-\frac {3}{4},2 i\right )+\operatorname {BesselJ}\left (\frac {1}{4},2 i\right )\right ) \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )+2 \left (2 \operatorname {BesselJ}\left (-\frac {3}{4},2 i\right )-i \operatorname {BesselJ}\left (\frac {1}{4},2 i\right )\right ) \left (\operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )-i x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )\right )}{x \left (\left (8 \operatorname {BesselJ}\left (-\frac {3}{4},2 i\right )-4 i \operatorname {BesselJ}\left (\frac {1}{4},2 i\right )\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )+\left (-4 \operatorname {BesselJ}\left (-\frac {5}{4},2 i\right )+5 i \operatorname {BesselJ}\left (-\frac {1}{4},2 i\right )+4 \operatorname {BesselJ}\left (\frac {3}{4},2 i\right )\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )\right )}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + y(x)**2 + Derivative(y(x), x),0)
ics = {y(-2): 1}
dsolve(ode,func=y(x),ics=ics)
TypeError : bad operand type for unary -: list