60.1.314 problem 315
Internal
problem
ID
[10328]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
315
Date
solved
:
Monday, January 27, 2025 at 07:13:01 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (2 x y^{3}-x^{4}\right ) y^{\prime }-y^{4}+2 x^{3} y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.028 (sec). Leaf size: 313
dsolve((2*x*y(x)^3-x^4)*diff(y(x),x)-y(x)^4+2*x^3*y(x) = 0,y(x), singsol=all)
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (x 12^{{1}/{3}} c_{1} +{\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}\right )}{6 c_{1} {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_{1} \right )}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
y &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) c_{1} \right ) 3^{{1}/{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.132 (sec). Leaf size: 54
DSolve[2*x^3*y[x] - y[x]^4 + (-x^4 + 2*x*y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {2 K[1]^3-1}{K[1] (K[1]+1) \left (K[1]^2-K[1]+1\right )}dK[1]=-\log (x)+c_1,y(x)\right ]
\]