28.1.47 problem 48
Internal
problem
ID
[4353]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
48
Date
solved
:
Monday, January 27, 2025 at 09:08:41 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 2 x^{2} y^{4}-y+\left (4 x^{3} y^{3}-x \right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.220 (sec). Leaf size: 2290
dsolve((2*x^2*y(x)^4-y(x))+(4*x^3*y(x)^3-x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+c_{1} \left (3^{{1}/{3}} 2^{{1}/{3}} x -3^{{5}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}} 2^{{2}/{3}}\right )\right ) c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}}}{6 c_{1} x} \\
y \left (x \right ) &= \frac {3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (2 \,3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+\left (3 \,2^{{2}/{3}} \left (i 3^{{1}/{3}}+\frac {3^{{5}/{6}}}{3}\right ) \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+x \left (i 3^{{5}/{6}} 2^{{1}/{3}}-6^{{1}/{3}}\right )\right ) c_{1} \right ) \left (i-\sqrt {3}\right )^{2} c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}}}{6 \left (1+i \sqrt {3}\right ) x c_{1}} \\
y \left (x \right ) &= \frac {3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (\sqrt {3}+i\right )^{2} \left (-2 \,3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+\left (3 \,2^{{2}/{3}} \left (i 3^{{1}/{3}}-\frac {3^{{5}/{6}}}{3}\right ) \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+2^{{1}/{3}} x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right ) c_{1} \right ) c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}}}{6 c_{1} x \left (i \sqrt {3}-1\right )} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (-3^{{5}/{6}} 2^{{2}/{3}} c_{1} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+6^{{1}/{3}} x c_{1} +3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}\right ) \left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) c_{1}^{2} x}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}}}{12 c_{1} x} \\
y \left (x \right ) &= \frac {3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+c_{1} \left (3^{{1}/{3}} 2^{{1}/{3}} x -3^{{5}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}} 2^{{2}/{3}}\right )\right ) c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12 c_{1} x} \\
y \left (x \right ) &= -\frac {3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (2 \,3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+\left (3 \,2^{{2}/{3}} \left (i 3^{{1}/{3}}+\frac {3^{{5}/{6}}}{3}\right ) \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+x \left (i 3^{{5}/{6}} 2^{{1}/{3}}-6^{{1}/{3}}\right )\right ) c_{1} \right ) \left (i-\sqrt {3}\right )^{2} c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}}}{12 x c_{1}} \\
y \left (x \right ) &= -\frac {3^{{1}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (2 \,3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+\left (3 \,2^{{2}/{3}} \left (i 3^{{1}/{3}}+\frac {3^{{5}/{6}}}{3}\right ) \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+x \left (i 3^{{5}/{6}} 2^{{1}/{3}}-6^{{1}/{3}}\right )\right ) c_{1} \right ) \left (i-\sqrt {3}\right )^{2} c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}} \left (3 \,3^{{1}/{6}}+i 3^{{2}/{3}}\right )}{12 x c_{1} \left (i-\sqrt {3}\right )} \\
y \left (x \right ) &= \frac {\left (-3 \,3^{{1}/{6}}+i 3^{{2}/{3}}\right ) {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (\sqrt {3}+i\right )^{2} \left (-2 \,3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+\left (3 \,2^{{2}/{3}} \left (i 3^{{1}/{3}}-\frac {3^{{5}/{6}}}{3}\right ) \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+\left (i 3^{{5}/{6}} 2^{{1}/{3}}+6^{{1}/{3}}\right ) x \right ) c_{1} \right ) c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}} 2^{{1}/{9}} 3^{{1}/{18}}}{12 c_{1} x \left (\sqrt {3}+i\right )} \\
y \left (x \right ) &= \frac {3^{{13}/{18}} 2^{{1}/{9}} {\left (\frac {\left (\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}}-\left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}\right ) x \left (\sqrt {3}+i\right )^{2} \left (-2 \,3^{{1}/{3}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{2}/{3}}+\left (3 \,2^{{2}/{3}} \left (i 3^{{1}/{3}}-\frac {3^{{5}/{6}}}{3}\right ) \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}+2^{{1}/{3}} x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right ) c_{1} \right ) c_{1}^{2}}{\left (\left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right )^{2} x^{2} c_{1}^{4}\right )^{{1}/{6}} \left (-x \left (-\sqrt {\frac {-2 x +27 c_{1}}{c_{1}}}+3 \sqrt {3}\right ) c_{1}^{2}\right )^{{1}/{3}}}\right )}^{{1}/{3}}}{12 c_{1} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 30.554 (sec). Leaf size: 318
DSolve[(2*x^2*y[x]^4-y[x])+(4*x^3*y[x]^3-x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {c_1 x}{\sqrt [3]{-54 x^4+6 \sqrt {3} \sqrt {-x^8 \left (-27+2 c_1{}^3 x\right )}}}+\frac {\sqrt [3]{-9 x^4+\sqrt {3} \sqrt {-x^8 \left (-27+2 c_1{}^3 x\right )}}}{6^{2/3} x^2} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-9 x^4+\sqrt {3} \sqrt {-x^8 \left (-27+2 c_1{}^3 x\right )}}}{2\ 6^{2/3} x^2}-\frac {i \left (\sqrt {3}-i\right ) c_1 x}{2 \sqrt [3]{-54 x^4+6 \sqrt {3} \sqrt {-x^8 \left (-27+2 c_1{}^3 x\right )}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) c_1 x}{2 \sqrt [3]{-54 x^4+6 \sqrt {3} \sqrt {-x^8 \left (-27+2 c_1{}^3 x\right )}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-9 x^4+\sqrt {3} \sqrt {-x^8 \left (-27+2 c_1{}^3 x\right )}}}{2\ 6^{2/3} x^2} \\
\end{align*}