73.8.25 problem 13.4 (f)
Internal
problem
ID
[15233]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
13.
Higher
order
equations:
Extending
first
order
concepts.
Additional
exercises
page
259
Problem
number
:
13.4
(f)
Date
solved
:
Tuesday, January 28, 2025 at 07:50:31 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} y^{2} y^{\prime \prime }+y^{\prime \prime }+2 y {y^{\prime }}^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.131 (sec). Leaf size: 293
dsolve(y(x)^2*diff(y(x),x$2)+diff(y(x),x$2)+2*y(x)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*}
y &= -i \\
y &= i \\
y &= \frac {\left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{2}/{3}}-4}{2 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{2}/{3}}+4 i \sqrt {3}-4}{4 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{2}/{3}}+4 i \sqrt {3}-\left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{2}/{3}}+4}{4 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.167 (sec). Leaf size: 307
DSolve[y[x]^2*D[y[x],{x,2}]+D[y[x],{x,2}]+2*y[x]*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-2+\sqrt [3]{2} \left (3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}} \\
y(x)\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}{2 \sqrt [3]{2}} \\
\end{align*}