15.6.7 problem 7
Internal
problem
ID
[2964]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
10,
page
41
Problem
number
:
7
Date
solved
:
Monday, January 27, 2025 at 07:04:37 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y+\left (2 x -3 y\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 305
dsolve(y(x)+(2*x-3*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {c_{1} \left (-4 x^{3}+27 c_{1} \right )}\right )^{{1}/{3}}}{6}+\frac {2 x^{2}}{3 \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {c_{1} \left (-4 x^{3}+27 c_{1} \right )}\right )^{{1}/{3}}}+\frac {x}{3} \\
y &= \frac {i \left (-\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{{2}/{3}}+4 x^{2}\right ) \sqrt {3}-{\left (\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{{1}/{3}}-2 x \right )}^{2}}{12 \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{{2}/{3}}-4 x^{2}\right ) \sqrt {3}-{\left (\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{{1}/{3}}-2 x \right )}^{2}}{12 \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.070 (sec). Leaf size: 379
DSolve[y[x]+(2*x-3*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{3} \left (\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}+\frac {x^2}{\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}}+x\right ) \\
y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-27 e^{c_1}}-\frac {2 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}}+4 x\right ) \\
y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-27 e^{c_1}}+\frac {2 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}}+4 x\right ) \\
\end{align*}