29.22.20 problem 628
Internal
problem
ID
[5220]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
628
Date
solved
:
Monday, January 27, 2025 at 10:22:58 AM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} 3 \left (x^{2}-y^{2}\right ) y^{\prime }+3 \,{\mathrm e}^{x}+6 x y \left (1+x \right )-2 y^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 341
dsolve(3*(x^2-y(x)^2)*diff(y(x),x)+3*exp(x)+6*x*y(x)*(1+x)-2*y(x)^3 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (2 \,{\mathrm e}^{4 x} x^{2}+2^{{1}/{3}} {\left (\left ({\mathrm e}^{3 x}+c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}}\right ) {\mathrm e}^{-2 x}}{2 {\left (\left ({\mathrm e}^{3 x}+c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {2^{{1}/{3}} {\mathrm e}^{-2 x} \left (-2 x^{2} {\mathrm e}^{4 x} \left (i \sqrt {3}-1\right )+2^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\left (\left ({\mathrm e}^{3 x}+c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}}\right )}{4 {\left (\left ({\mathrm e}^{3 x}+c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (-2 x^{2} {\mathrm e}^{4 x} \left (1+i \sqrt {3}\right )+2^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\left (\left ({\mathrm e}^{3 x}+c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}}\right ) 2^{{1}/{3}} {\mathrm e}^{-2 x}}{4 {\left (\left ({\mathrm e}^{3 x}+c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.300 (sec). Leaf size: 497
DSolve[3(x^2-y[x]^2)D[y[x],x]+3 Exp[x]+6 x y[x](1+x)-2 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} e^{2 x} x^2}{\sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{2 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) e^{2 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{2 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) e^{2 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\
\end{align*}