44.4.35 problem 16
Internal
problem
ID
[7048]
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
:
16
Date
solved
:
Wednesday, March 05, 2025 at 04:04:00 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x \left (y-4\right )^{2}-2 \end{align*}
✓ Maple. Time used: 0.023 (sec). Leaf size: 113
ode:=diff(y(x),x) = x*(y(x)-4)^2-2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 \left (2^{{1}/{3}} \operatorname {AiryAi}\left (1, -\frac {8 \,2^{{2}/{3}} x}{\left (-32 i\right )^{{2}/{3}}}\right ) \left (1+i \sqrt {3}\right ) c_{1} +2^{{1}/{3}} \operatorname {AiryBi}\left (1, -\frac {8 \,2^{{2}/{3}} x}{\left (-32 i\right )^{{2}/{3}}}\right ) \left (1+i \sqrt {3}\right )+\operatorname {AiryAi}\left (-\frac {8 \,2^{{2}/{3}} x}{\left (-32 i\right )^{{2}/{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (-\frac {8 \,2^{{2}/{3}} x}{\left (-32 i\right )^{{2}/{3}}}\right )\right ) 2^{{2}/{3}}}{\left (1+i \sqrt {3}\right ) \left (c_{1} \operatorname {AiryAi}\left (1, -\frac {8 \,2^{{2}/{3}} x}{\left (-32 i\right )^{{2}/{3}}}\right )+\operatorname {AiryBi}\left (1, -\frac {8 \,2^{{2}/{3}} x}{\left (-32 i\right )^{{2}/{3}}}\right )\right )}
\]
✓ Mathematica. Time used: 0.192 (sec). Leaf size: 113
ode=D[y[x],x]==x*(y[x]-4)^2-2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {2^{2/3} \operatorname {AiryBi}\left (\sqrt [3]{2} x\right )-4 \operatorname {AiryBiPrime}\left (\sqrt [3]{2} x\right )+2^{2/3} c_1 \operatorname {AiryAi}\left (\sqrt [3]{2} x\right )-4 c_1 \operatorname {AiryAiPrime}\left (\sqrt [3]{2} x\right )}{\operatorname {AiryBiPrime}\left (\sqrt [3]{2} x\right )+c_1 \operatorname {AiryAiPrime}\left (\sqrt [3]{2} x\right )} \\
y(x)\to 4-\frac {2^{2/3} \operatorname {AiryAi}\left (\sqrt [3]{2} x\right )}{\operatorname {AiryAiPrime}\left (\sqrt [3]{2} x\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*(y(x) - 4)**2 + Derivative(y(x), x) + 2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Rational Solution doesnt exist