64.7.4 problem 4
Internal
problem
ID
[13378]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
2,
Section
2.4.
Special
integrating
factors
and
transformations.
Exercises
page
67
Problem
number
:
4
Date
solved
:
Tuesday, January 28, 2025 at 05:36:08 AM
CAS
classification
:
[_rational]
\begin{align*} 2 x y^{2}+y+\left (2 y^{3}-x \right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 297
dsolve((2*x*y(x)^2+y(x))+(2*y(x)^3-x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {-12 x^{2}-12 c_{1} +\left (-108 x +12 \sqrt {12 x^{6}+36 c_{1} x^{4}+\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{{2}/{3}}}{6 \left (-108 x +12 \sqrt {12 x^{6}+36 c_{1} x^{4}+\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108 x +12 \sqrt {12 x^{6}+36 c_{1} x^{4}+\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{{2}/{3}}+\left (x^{2}+c_{1} \right ) \left (i \sqrt {3}-1\right )}{\left (-108 x +12 \sqrt {12 x^{6}+36 c_{1} x^{4}+\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-108 x +12 \sqrt {12 x^{6}+36 c_{1} x^{4}+\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{{2}/{3}}}{12}+\left (x^{2}+c_{1} \right ) \left (1+i \sqrt {3}\right )}{\left (-108 x +12 \sqrt {12 x^{6}+36 c_{1} x^{4}+\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.144 (sec). Leaf size: 316
DSolve[(2*x*y[x]^2+y[x])+(2*y[x]^3-x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2^{2/3} \left (-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}\right ){}^{2/3}-6 \sqrt [3]{2} \left (x^2-c_1\right )}{6 \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^2-c_1\right )}{2^{2/3} \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}{6 \sqrt [3]{2}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^2-c_1\right )}{2^{2/3} \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}{6 \sqrt [3]{2}} \\
y(x)\to 0 \\
\end{align*}