44.3.25 problem 33
Internal
problem
ID
[7006]
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
:
33
Date
solved
:
Monday, January 27, 2025 at 02:40:44 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} y^{\prime \prime }&=2 y {y^{\prime }}^{3} \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 320
dsolve(diff(y(x),x$2)=2*y(x)*diff(y(x),x)^3,y(x), singsol=all)
\begin{align*}
y &= c_{1} \\
y &= \frac {\left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{2}/{3}}+4 c_{1}}{2 \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {-i \sqrt {3}\, \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, c_{1} -\left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{2}/{3}}-4 c_{1}}{4 \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {-i \sqrt {3}\, \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, c_{1} +\left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{2}/{3}}+4 c_{1}}{4 \left (-12 c_{2} -12 x +4 \sqrt {-4 c_{1}^{3}+9 c_{2}^{2}+18 c_{2} x +9 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.345 (sec). Leaf size: 351
DSolve[D[y[x],{x,2}]==2*y[x]*D[y[x],x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}-\frac {\sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}{\sqrt [3]{2}} \\
y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2\right ){}^{2/3}+\sqrt [3]{2} \left (-2-2 i \sqrt {3}\right ) c_1}{4 \sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}} \\
y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2\right ){}^{2/3}+2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) c_1}{4 \sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}} \\
y(x)\to 0 \\
\end{align*}