44.2.21 problem 21
Internal
problem
ID
[6953]
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
:
Monday, January 27, 2025 at 02:38:15 PM
CAS
classification
:
[_separable]
\begin{align*} \left (4-y^{2}\right ) y^{\prime }&=x^{2} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 234
dsolve((4-y(x)^2)*diff(y(x),x)=x^2,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*}
✓ Solution by Mathematica
Time used: 2.550 (sec). Leaf size: 321
DSolve[(4-y[x]^2)*D[y[x],x]==x^2,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*}