82.5.1 problem Ex. 1
Internal
problem
ID
[18737]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
II.
Equations
of
the
first
order
and
of
the
first
degree.
Exercises
at
page
20
Problem
number
:
Ex.
1
Date
solved
:
Tuesday, January 28, 2025 at 12:13:50 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} x^{2}-4 y x -2 y^{2}+\left (y^{2}-4 y x -2 x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 436
dsolve((x^2-4*x*y(x)-2*y(x)^2)+(y(x)^2-4*x*y(x)-2*x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{1}/{3}}}{2}+\frac {12 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{1}/{3}}}+2 c_{1} x}{c_{1}} \\
y \left (x \right ) &= \frac {-\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{1}/{3}}}{4}-\frac {6 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{1}/{3}}}+2 c_{1} x -\frac {i \sqrt {3}\, \left (-24 x^{2} c_{1}^{2}+\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{2}/{3}}\right )}{4 \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{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 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{2}/{3}}+24 x^{2} c_{1}^{2}-8 c_{1} x \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{1}/{3}}+\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{2}/{3}}}{4 \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 30.810 (sec). Leaf size: 781
DSolve[(x^2-4*x*y[x]-2*y[x]^2)+(y[x]^2-4*x*y[x]-2*x^2)*D[y[x],x]==0,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*}