60.1.282 problem 283
Internal
problem
ID
[10296]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
283
Date
solved
:
Monday, January 27, 2025 at 06:51:02 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} 3 \left (y^{2}-x^{2}\right ) y^{\prime }+2 y^{3}-6 x \left (1+x \right ) y-3 \,{\mathrm e}^{x}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 353
dsolve(3*(y(x)^2-x^2)*diff(y(x),x)+2*y(x)^3-6*x*(x+1)*y(x)-3*exp(x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {2^{{1}/{3}} \left (2 x^{2} {\mathrm e}^{4 x}+2^{{1}/{3}} {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 \,{\mathrm e}^{3 x} c_{1} +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 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
y &= -\frac {2^{{1}/{3}} {\mathrm e}^{-2 x} \left (-2 \left (i \sqrt {3}-1\right ) x^{2} {\mathrm e}^{4 x}+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 \,{\mathrm e}^{3 x} c_{1} +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 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
y &= \frac {2^{{1}/{3}} \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 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}}\right ) {\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 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.264 (sec). Leaf size: 497
DSolve[3*(y[x]^2-x^2)*D[y[x],x]+2*y[x]^3-6*x*(x+1)*y[x]-3*Exp[x]==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*}