38.2.21 problem 21
Internal
problem
ID
[8239]
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.
Section
1.2
Initial
value
problems.
Exercises
1.2
at
page
19
Problem
number
:
21
Date
solved
:
Tuesday, September 30, 2025 at 05:19:41 PM
CAS
classification
:
[_separable]
\begin{align*} \left (4-y^{2}\right ) y^{\prime }&=x^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 234
ode:=(4-y(x)^2)*diff(y(x),x) = x^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}-256}\right )^{{2}/{3}}+16}{2 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}-256}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}-256}\right )^{{2}/{3}}-16 i \sqrt {3}+16}{4 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}-256}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}-256}\right )^{{2}/{3}}-16 i \sqrt {3}-16}{4 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+6 c_1 \,x^{3}+9 c_1^{2}-256}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 1.942 (sec). Leaf size: 321
ode=(4-y[x]^2)*D[y[x],x]==x^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {8+\sqrt [3]{2} \left (-x^3+\sqrt {x^6-6 c_1 x^3-256+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-256+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-256+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-256+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{\sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-256+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-256+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \end{align*}
✓ Sympy. Time used: 14.195 (sec). Leaf size: 338
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + (4 - y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}{4} + \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}{4} - \frac {8}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}{4} - \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}{4} - \frac {8}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}{2} - \frac {4}{\sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 256}}}\right )\right ]
\]