28.1.109 problem 132
Internal
problem
ID
[4415]
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
:
132
Date
solved
:
Monday, January 27, 2025 at 09:15:15 AM
CAS
classification
:
[_rational]
\begin{align*} y+3 x^{4} y^{2}+\left (x +2 x^{2} y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 357
dsolve((y(x)+3*x^4*y(x)^2)+(x+2*x^2*y(x)^3)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {12^{{1}/{3}} \left (-{\left (\left (9+\sqrt {12 x^{11}-36 c_{1} x^{8}+36 c_{1}^{2} x^{5}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+12^{{1}/{3}} x^{2} \left (x^{3}-c_{1} \right )\right )}{6 {\left (\left (9+\sqrt {12 x^{11}-36 c_{1} x^{8}+36 c_{1}^{2} x^{5}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (9+\sqrt {12 x^{11}-36 c_{1} x^{8}+36 c_{1}^{2} x^{5}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) \left (x^{3}-c_{1} \right )\right )}{12 {\left (\left (9+\sqrt {12 x^{11}-36 c_{1} x^{8}+36 c_{1}^{2} x^{5}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (9+\sqrt {12 x^{11}-36 c_{1} x^{8}+36 c_{1}^{2} x^{5}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{2} \left (x^{3}-c_{1} \right ) \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right )}{12 {\left (\left (9+\sqrt {12 x^{11}-36 c_{1} x^{8}+36 c_{1}^{2} x^{5}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.121 (sec). Leaf size: 358
DSolve[(y[x]+3*x^4*y[x]^2)+(x+2*x^2*y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (-x^4+c_1 x\right )}{\sqrt [3]{27 x^2+\sqrt {729 x^4+108 x^3 \left (x^4-c_1 x\right ){}^3}}}+\frac {\sqrt [3]{27 x^2+\sqrt {729 x^4+108 x^3 \left (x^4-c_1 x\right ){}^3}}}{3 \sqrt [3]{2} x} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^4-c_1 x\right )}{2^{2/3} \sqrt [3]{27 x^2+\sqrt {729 x^4+108 x^3 \left (x^4-c_1 x\right ){}^3}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^2+\sqrt {729 x^4+108 x^3 \left (x^4-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^4-c_1 x\right )}{2^{2/3} \sqrt [3]{27 x^2+\sqrt {729 x^4+108 x^3 \left (x^4-c_1 x\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^2+\sqrt {729 x^4+108 x^3 \left (x^4-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x} \\
\end{align*}