15.7.2 problem 2
Internal
problem
ID
[2983]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
11,
page
45
Problem
number
:
2
Date
solved
:
Tuesday, March 04, 2025 at 03:37:15 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{3} y^{\prime }+x y^{4}&=x \,{\mathrm e}^{-x^{2}} \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 112
ode:=y(x)^3*diff(y(x),x)+x*y(x)^4 = x*exp(-x^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= {\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_{1} \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\
y &= -{\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_{1} \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\
y &= -i {\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_{1} \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\
y &= i {\mathrm e}^{-x^{2}} {\left (\left (2 \,{\mathrm e}^{x^{2}}+c_{1} \right ) {\mathrm e}^{2 x^{2}}\right )}^{{1}/{4}} \\
\end{align*}
✓ Mathematica. Time used: 0.414 (sec). Leaf size: 200
ode=y[x]^3*D[y[x],x]+x*y[x]^4==x*exp[-x^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1} \\
y(x)\to -i e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1} \\
y(x)\to i e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1} \\
y(x)\to e^{-\frac {x^2}{2}} \sqrt [4]{4 \int _1^xe^{2 K[1]^2} \exp \left (-K[1]^2\right ) K[1]dK[1]+c_1} \\
\end{align*}
✓ Sympy. Time used: 2.822 (sec). Leaf size: 82
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*y(x)**4 - x*exp(-x**2) + y(x)**3*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - i \sqrt [4]{\left (C_{1} e^{- x^{2}} + 2\right ) e^{- x^{2}}}, \ y{\left (x \right )} = i \sqrt [4]{\left (C_{1} e^{- x^{2}} + 2\right ) e^{- x^{2}}}, \ y{\left (x \right )} = - \sqrt [4]{\left (C_{1} e^{- x^{2}} + 2\right ) e^{- x^{2}}}, \ y{\left (x \right )} = \sqrt [4]{\left (C_{1} e^{- x^{2}} + 2\right ) e^{- x^{2}}}\right ]
\]