7.5.51 problem 51
Internal
problem
ID
[155]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.6
(substitution
and
exact
equations).
Problems
at
page
72
Problem
number
:
51
Date
solved
:
Friday, February 07, 2025 at 07:58:55 AM
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: 0.013 (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 \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}} \sqrt {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 \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}} \sqrt {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: 9.378 (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*}