29.22.18 problem 626
Internal
problem
ID
[5218]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
626
Date
solved
:
Monday, January 27, 2025 at 10:22:50 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x^{2}-3 y^{2}\right ) y^{\prime }+1+2 y x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 294
dsolve((x^2-3*y(x)^2)*diff(y(x),x)+1+2*x*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{2}/{3}}+12 x^{2}}{6 \left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {12 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{2}/{3}}-12 x^{2}-\left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{2}/{3}}}{12 \left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{1}/{3}}}{12}-\frac {x^{2} \left (1+i \sqrt {3}\right )}{\left (108 x +108 c_{1} +12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.796 (sec). Leaf size: 307
DSolve[(x^2-3 y[x]^2)D[y[x],x]+1+2 x y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\
\end{align*}