83.8.13 problem 14
Internal
problem
ID
[19139]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Misc
examples
on
chapter
II
at
page
25
Problem
number
:
14
Date
solved
:
Tuesday, January 28, 2025 at 01:05:13 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y \left (2 x^{2} y+{\mathrm e}^{x}\right )-\left ({\mathrm e}^{x}+y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 314
dsolve(y(x)*(2*x^2*y(x)+exp(x))-(exp(x)+y(x)^3)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {4 x^{3}+\left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_{1} x^{6}-432 x^{3} c_{1}^{2}-216 c_{1}^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{2}/{3}}+6 c_{1}}{3 \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_{1} x^{6}-432 x^{3} c_{1}^{2}-216 c_{1}^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_{1} x^{6}-432 x^{3} c_{1}^{2}-216 c_{1}^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}}{6}+\frac {2 \left (x^{3}+\frac {3 c_{1}}{2}\right ) \left (i \sqrt {3}-1\right )}{3 \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_{1} x^{6}-432 x^{3} c_{1}^{2}-216 c_{1}^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_{1} x^{6}-432 x^{3} c_{1}^{2}-216 c_{1}^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{2}/{3}}}{6}+\frac {2 \left (-1-i \sqrt {3}\right ) \left (x^{3}+\frac {3 c_{1}}{2}\right )}{3}}{\left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_{1} x^{6}-432 x^{3} c_{1}^{2}-216 c_{1}^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 21.616 (sec). Leaf size: 336
DSolve[y[x]*(2*x^2*y[x]+Exp[x])-(Exp[x]+y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {4 x^3+\left (27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}\right ){}^{2/3}+6 c_1}{3 \sqrt [3]{27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}}} \\
y(x)\to -\frac {i \left (-4 \left (\sqrt {3}+i\right ) x^3+\left (\sqrt {3}-i\right ) \left (27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}\right ){}^{2/3}-6 \left (\sqrt {3}+i\right ) c_1\right )}{6 \sqrt [3]{27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}}} \\
y(x)\to \frac {i \left (-4 \left (\sqrt {3}-i\right ) x^3+\left (\sqrt {3}+i\right ) \left (27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}\right ){}^{2/3}-6 \left (\sqrt {3}-i\right ) c_1\right )}{6 \sqrt [3]{27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}}} \\
y(x)\to 0 \\
\end{align*}