15.3.2 problem 2
Internal
problem
ID
[2895]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
7,
page
28
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 06:27:45 AM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]
\begin{align*} x +\left (x -2 y+2\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.970 (sec). Leaf size: 149
dsolve(x+(x-2*y(x)+2)*diff(y(x),x)=0,y(x), singsol=all)
\[
y = -\frac {-2 \left (2 c_1 \,x^{3}+2 \sqrt {-2 \left (c_1 \,x^{3}-\frac {1}{2}\right ) c_1^{2} x^{6}}\right )^{{2}/{3}} c_1 \,x^{2}-\frac {\left (1+i \sqrt {3}\right ) \left (c_1 \,x^{3}+\sqrt {-2 \left (c_1 \,x^{3}-\frac {1}{2}\right ) c_1^{2} x^{6}}\right ) \left (2 c_1 \,x^{3}+2 \sqrt {-2 \left (c_1 \,x^{3}-\frac {1}{2}\right ) c_1^{2} x^{6}}\right )^{{1}/{3}}}{2}+x^{6} \left (i \sqrt {3}-1\right ) c_1^{2}}{2 \left (2 c_1 \,x^{3}+2 \sqrt {-2 \left (c_1 \,x^{3}-\frac {1}{2}\right ) c_1^{2} x^{6}}\right )^{{2}/{3}} x^{2} c_1}
\]
✓ Solution by Mathematica
Time used: 60.039 (sec). Leaf size: 445
DSolve[x+(x-2*y[x]+2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,1\right ]} \\
y(x)\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,2\right ]} \\
y(x)\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,3\right ]} \\
y(x)\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,4\right ]} \\
y(x)\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,5\right ]} \\
y(x)\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,6\right ]} \\
\end{align*}