29.21.26 problem 602
Internal
problem
ID
[5196]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
21
Problem
number
:
602
Date
solved
:
Monday, January 27, 2025 at 10:20:30 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 317
dsolve((x^2+y(x)^2)*diff(y(x),x)+2*x*(2*x+y(x)) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {x^{2} \left (i \sqrt {3}-1\right ) \sqrt {c_{1}}}{\left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4-16 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 53.999 (sec). Leaf size: 576
DSolve[(x^2+y[x]^2)D[y[x],x]+2 x(2 x+y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\
y(x)\to \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}-\frac {x^2}{\sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \\
y(x)\to \frac {\sqrt [3]{-1} \left (x^2+\sqrt [3]{-1} \left (\sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}\right )}{\sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2+\left (-1-i \sqrt {3}\right ) \left (\sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \\
\end{align*}