36.4.3 problem 3
Internal
problem
ID
[6341]
Book
:
Fundamentals
of
Differential
Equations.
By
Nagle,
Saff
and
Snider.
9th
edition.
Boston.
Pearson
2018.
Section
:
Chapter
2,
First
order
differential
equations.
Review
problems.
page
79
Problem
number
:
3
Date
solved
:
Monday, January 27, 2025 at 01:57:35 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x^{2}-\frac {2}{y^{3}}\right ) y^{\prime }+2 y x -3 x^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 689
dsolve((x^2-2*y(x)^(-3))*diff(y(x),x)+(2*x*y(x)-3*x^2)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\frac {\left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{1}/{3}}}{2}+\frac {2 \left (-x^{3}+c_1 \right )^{2}}{\left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{1}/{3}}}+x^{3}-c_1}{3 x^{2}} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{2}/{3}}}{4}+\left (\left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{1}/{3}}+\left (x^{3}-c_1 \right ) \left (i \sqrt {3}-1\right )\right ) \left (x^{3}-c_1 \right )}{3 \left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{1}/{3}} x^{2}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{2}/{3}}}{4}+\left (x^{3}-c_1 \right ) \left (\left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{1}/{3}}+\left (-i \sqrt {3}-1\right ) \left (x^{3}-c_1 \right )\right )}{3 \left (8 x^{9}-24 c_1 \,x^{6}+24 c_1^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_1 \,x^{6}-12 c_1^{2} x^{3}+27 x^{4}+4 c_1^{3}}\, x^{2}-108 x^{4}-8 c_1^{3}\right )^{{1}/{3}} x^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 22.281 (sec). Leaf size: 676
DSolve[(x^2-2*y[x]^(-3))*D[y[x],x]+(2*x*y[x]-3*x^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2 \left (x^3+c_1\right )+\frac {2 \left (x^3+c_1\right ){}^2}{\sqrt [3]{x^9+3 c_1 x^6-\frac {27 x^4}{2}+3 c_1{}^2 x^3+\frac {3}{2} \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+c_1{}^3}}+2^{2/3} \sqrt [3]{2 x^9+6 c_1 x^6-27 x^4+6 c_1{}^2 x^3+3 \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+2 c_1{}^3}}{6 x^2} \\
y(x)\to \frac {4 \left (x^3+c_1\right )-\frac {2 i \left (\sqrt {3}-i\right ) \left (x^3+c_1\right ){}^2}{\sqrt [3]{x^9+3 c_1 x^6-\frac {27 x^4}{2}+3 c_1{}^2 x^3+\frac {3}{2} \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^9+6 c_1 x^6-27 x^4+6 c_1{}^2 x^3+3 \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+2 c_1{}^3}}{12 x^2} \\
y(x)\to \frac {4 \left (x^3+c_1\right )+\frac {2 i \left (\sqrt {3}+i\right ) \left (x^3+c_1\right ){}^2}{\sqrt [3]{x^9+3 c_1 x^6-\frac {27 x^4}{2}+3 c_1{}^2 x^3+\frac {3}{2} \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^9+6 c_1 x^6-27 x^4+6 c_1{}^2 x^3+3 \sqrt {3} \sqrt {-x^4 \left (4 x^9+12 c_1 x^6-27 x^4+12 c_1{}^2 x^3+4 c_1{}^3\right )}+2 c_1{}^3}}{12 x^2} \\
y(x)\to 0 \\
\end{align*}