29.35.27 problem 1061
Internal
problem
ID
[5625]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1061
Date
solved
:
Monday, January 27, 2025 at 12:29:43 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: 0.263 (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 \left (x \right ) &= \frac {3 x^{{4}/{3}}}{2} \\
y \left (x \right ) &= -\frac {3 x^{{4}/{3}} \left (1+i \sqrt {3}\right )}{4} \\
y \left (x \right ) &= \frac {3 x^{{4}/{3}} \left (i \sqrt {3}-1\right )}{4} \\
y \left (x \right ) &= \frac {c_{1}^{3}-128 x^{2}}{32 c_{1}} \\
y \left (x \right ) &= \frac {c_{1}^{3}+128 x^{2}}{32 c_{1}} \\
y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{{1}/{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{{1}/{3}}}+\frac {c_{1}^{2}}{96} \\
y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{{1}/{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{{1}/{3}}}+\frac {c_{1}^{2}}{96} \\
y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right )}{192 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}} c_{1}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}}\right )}{192 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}} c_{1}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 153.799 (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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