78.8.10 problem 10
Internal
problem
ID
[18208]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Miscellaneous
Problems
for
Chapter
2.
Problems
at
page
99
Problem
number
:
10
Date
solved
:
Tuesday, January 28, 2025 at 11:39:00 AM
CAS
classification
:
[_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} \left ({\mathrm e}^{x}-3 x^{2} y^{2}\right ) y^{\prime }+y \,{\mathrm e}^{x}&=2 x y^{3} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 274
dsolve((exp(x)-3*x^2*y(x)^2)*diff(y(x),x)+y(x)*exp(x)=2*x*y(x)^3,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}+12 \,{\mathrm e}^{x}}{6 \left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{1}/{3}} x} \\
y &= \frac {-i \sqrt {3}\, \left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}+12 i \sqrt {3}\, {\mathrm e}^{x}-\left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}-12 \,{\mathrm e}^{x}}{12 \left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{1}/{3}} x} \\
y &= -\frac {-i \sqrt {3}\, \left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}+12 i \sqrt {3}\, {\mathrm e}^{x}+\left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}+12 \,{\mathrm e}^{x}}{12 \left (108 c_{1} x +12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.255 (sec). Leaf size: 364
DSolve[(Exp[x]-3*x^2*y[x]^2)*D[y[x],x]+y[x]*Exp[x]==2*x*y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2 \sqrt [3]{3} e^x x^2+\sqrt [3]{2} \left (9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}\right ){}^{2/3}}{6^{2/3} x^2 \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}}{2 \sqrt [3]{2} 3^{2/3} x^2}-\frac {\left (\sqrt {3}+3 i\right ) e^x}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\
y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}}{2 \sqrt [3]{2} 3^{2/3} x^2}-\frac {\left (\sqrt {3}-3 i\right ) e^x}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\
\end{align*}