12.5.36 problem 33
Internal
problem
ID
[1660]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Transformation
of
Nonlinear
Equations
into
Separable
Equations.
Section
2.4
Page
68
Problem
number
:
33
Date
solved
:
Monday, January 27, 2025 at 05:19:27 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{\prime }&=\frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}} \end{align*}
✓ Solution by Maple
Time used: 2.000 (sec). Leaf size: 129
dsolve(diff(y(x),x)=(x*y(x)^2+2*y(x)^3)/(x^3+x^2*y(x)+x*y(x)^2),y(x), singsol=all)
\[
y = x \left (\operatorname {RootOf}\left (x^{2} c_1 \,\textit {\_Z}^{8}+2 x^{2} c_1 \,\textit {\_Z}^{6}+c_1 \,\textit {\_Z}^{4} x^{2}-2 \textit {\_Z}^{2}-1\right )^{6} c_1 \,x^{2}+2 \operatorname {RootOf}\left (x^{2} c_1 \,\textit {\_Z}^{8}+2 x^{2} c_1 \,\textit {\_Z}^{6}+c_1 \,\textit {\_Z}^{4} x^{2}-2 \textit {\_Z}^{2}-1\right )^{4} c_1 \,x^{2}+\operatorname {RootOf}\left (x^{2} c_1 \,\textit {\_Z}^{8}+2 x^{2} c_1 \,\textit {\_Z}^{6}+c_1 \,\textit {\_Z}^{4} x^{2}-2 \textit {\_Z}^{2}-1\right )^{2} c_1 \,x^{2}-1\right )
\]
✓ Solution by Mathematica
Time used: 60.170 (sec). Leaf size: 1989
DSolve[D[y[x],x]==(x*y[x]^2+2*y[x]^3)/(x^3+x^2*y[x]+x*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}+\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\
y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}+\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\
y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}-\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\
y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}-\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\
\end{align*}