54.2.24 problem 27
Internal
problem
ID
[8556]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
16.
Nonlinear
equations.
Miscellaneous
Exercises.
Page
340
Problem
number
:
27
Date
solved
:
Monday, January 27, 2025 at 04:14:12 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}+4 x^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.214 (sec). Leaf size: 792
dsolve(x*diff(y(x),x)^3-2*y(x)*diff(y(x),x)^2+4*x^2=0,y(x), singsol=all)
\begin{align*}
y &= \frac {3 x^{{4}/{3}}}{2} \\
y &= -\frac {3 x^{{4}/{3}} \left (1+i \sqrt {3}\right )}{4} \\
y &= \frac {3 x^{{4}/{3}} \left (i \sqrt {3}-1\right )}{4} \\
y &= \frac {c_{1}^{3}-128 x^{2}}{32 c_{1}} \\
y &= \frac {c_{1}^{3}+128 x^{2}}{32 c_{1}} \\
y &= \frac {c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{{1}/{3}}}{96}+\frac {c_{1}^{3}}{96 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{{1}/{3}}}+\frac {c_{1}^{2}}{96} \\
y &= \frac {c_{1} \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}\right )^{{1}/{3}}}{96}+\frac {c_{1}^{3}}{96 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}\right )^{{1}/{3}}}+\frac {c_{1}^{2}}{96} \\
y &= \frac {\left (c_{1} -\left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right ) c_{1} \left (i \left (\left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right )}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right ) c_{1}^{2}}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}} \\
y &= \frac {\left (c_{1} -\left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right ) c_{1} \left (i \left (\left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right )}{192 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) c_{1} \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right ) c_{1}^{2}}{192 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 111.655 (sec). Leaf size: 15120
DSolve[x*D[y[x],x]^3-2*y[x]*D[y[x],x]^2+4*x^2==0,y[x],x,IncludeSingularSolutions -> True]
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