30.1.6 problem Example, page 36
Internal
problem
ID
[5694]
Book
:
Differential
and
integral
calculus,
vol
II
By
N.
Piskunov.
1974
Section
:
Chapter
1
Problem
number
:
Example,
page
36
Date
solved
:
Monday, January 27, 2025 at 01:07:25 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 313
dsolve(2*x/y(x)^3+ (y(x)^2-3*x^2)/(y(x)^4)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {1+\frac {\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{1}/{3}}}{2}+\frac {2}{\left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{1}/{3}}}}{3 c_{1}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{1}/{3}}+4}{12 \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{1}/{3}} c_{1}} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{1}/{3}}-4}{12 \left (-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, x \sqrt {27 c_{1}^{2} x^{2}-4}\, c_{1} +8\right )^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.202 (sec). Leaf size: 458
DSolve[2*x/y[x]^3+(y[x]^2-3*x^2)/(y[x]^4)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right ) \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
\end{align*}