29.21.25 problem 601
Internal
problem
ID
[5195]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
21
Problem
number
:
601
Date
solved
:
Monday, January 27, 2025 at 10:18:55 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (x^{2}-y^{2}\right ) y^{\prime }+x \left (x +2 y\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 316
dsolve((x^2-y(x)^2)*diff(y(x),x)+x*(x+2*y(x)) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {4 c_{1} x^{2}+\left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{2 \left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\
y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}-i \left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}} \sqrt {3}-4 c_{1} x^{2}-\left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {x^{2} \sqrt {c_{1}}\, \left (1+i \sqrt {3}\right )}{\left (4+4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {-3 x^{6} c_{1}^{3}+2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.199 (sec). Leaf size: 359
DSolve[(x^2-y[x]^2)*D[y[x],x]+x*(x+2*y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{x^3+\sqrt {-3 x^6+2 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]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {i \left (\sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}-2 \left (\sqrt {3}-i\right ) x^2\right )}{2\ 2^{2/3} \sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2+2^{2/3} \left (-1-i \sqrt {3}\right ) \left (x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
\end{align*}