15.5.2 problem 2
Internal
problem
ID
[2938]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
9,
page
38
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 06:59:42 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y x +\left (y+x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.411 (sec). Leaf size: 973
dsolve(x*y(x)+(x^2+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-1+\frac {\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}\right ) x^{2}}{2} \\
y &= \frac {\left (-1+\frac {\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}\right ) x^{2}}{2} \\
y &= \frac {\left (-1+\frac {\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}\right ) x^{2}}{2} \\
y &= \frac {i \sqrt {3}\, x^{4}-i \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{2}/{3}} \sqrt {3}-x^{4}-2 x^{2} \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}-\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{2}/{3}}}{4 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {x^{2} \left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}\right )}{4 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, x^{4}-i \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{2}/{3}} \sqrt {3}-x^{4}-2 x^{2} \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}-\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{2}/{3}}}{4 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {x^{2} \left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}\right )}{4 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, x^{4}-i \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{2}/{3}} \sqrt {3}-x^{4}-2 x^{2} \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}-\left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{2}/{3}}}{4 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {x^{2} \left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}\right )}{4 \left (2 c_1^{2}-x^{6}+2 c_1 \sqrt {-x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.040 (sec). Leaf size: 397
DSolve[x*y[x]+(x^2+y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,1\right ]} \\
y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,2\right ]} \\
y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,3\right ]} \\
y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,4\right ]} \\
y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,5\right ]} \\
y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,6\right ]} \\
\end{align*}