29.22.13 problem 621
Internal
problem
ID
[5213]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
621
Date
solved
:
Monday, January 27, 2025 at 10:21:23 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (2 x^{2}+4 y x -y^{2}\right ) y^{\prime }&=x^{2}-4 y x -2 y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.058 (sec). Leaf size: 436
dsolve((2*x^2+4*x*y(x)-y(x)^2)*diff(y(x),x) = x^2-4*x*y(x)-2*y(x)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\frac {\left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}{2}+\frac {12 x^{2} c_{1}^{2}}{\left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}+2 c_{1} x}{c_{1}} \\
y \left (x \right ) &= \frac {-\frac {\left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}{4}-\frac {6 x^{2} c_{1}^{2}}{\left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}+2 c_{1} x -\frac {i \sqrt {3}\, \left (-24 c_{1}^{2} x^{2}+\left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}\right )}{4 \left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}}{c_{1}} \\
y \left (x \right ) &= -\frac {24 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}+24 c_{1}^{2} x^{2}-8 c_{1} x \left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}+\left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}}{4 \left (108 c_{1}^{3} x^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 31.079 (sec). Leaf size: 781
DSolve[(2 x^2+4 x y[x]-y[x]^2)D[y[x],x]==x^2-4 x y[x]-2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {6 \sqrt [3]{2} x^2}{\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\
y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\
y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{27 x^3+\sqrt {-135 x^6+54 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\
y(x)\to \frac {4 \sqrt [3]{2} 3^{2/3} x^2+4 \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3} x+2^{2/3} \sqrt [3]{3} \left (\sqrt {15} \sqrt {-x^6}+9 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}} \\
y(x)\to \frac {-4 \sqrt [3]{2} 3^{2/3} x^2+12 i \sqrt [3]{2} \sqrt [6]{3} x^2+8 \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3} x-i 2^{2/3} 3^{5/6} \left (\sqrt {15} \sqrt {-x^6}+9 x^3\right )^{2/3}-2^{2/3} \sqrt [3]{3} \left (\sqrt {15} \sqrt {-x^6}+9 x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}} \\
y(x)\to \frac {\sqrt [3]{3} \left (\sqrt {15} \sqrt {-x^6}+9 x^3\right )^{2/3} \text {Root}\left [2 \text {$\#$1}^3-1\&,3\right ]-2 \sqrt [3]{-2} 3^{2/3} x^2+2 \sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3} x}{\sqrt [3]{\sqrt {15} \sqrt {-x^6}+9 x^3}} \\
\end{align*}