10.5.18 problem 25
Internal
problem
ID
[1210]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Section
2.6.
Page
100
Problem
number
:
25
Date
solved
:
Saturday, March 29, 2025 at 10:47:28 PM
CAS
classification
:
[[_homogeneous, `class D`], _rational]
\begin{align*} 2 x y+3 x^{2} y+y^{3}+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.095 (sec). Leaf size: 293
ode:=2*x*y(x)+3*x^2*y(x)+y(x)^3+(x^2+y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {2^{{1}/{3}} {\mathrm e}^{-3 x} \left (x^{2} {\mathrm e}^{6 x} c_1^{2}-\frac {2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}}}{2}\right )}{{\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= -\frac {\left (\frac {{\mathrm e}^{-3 x} 2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}} \left (1+i \sqrt {3}\right )}{2}+{\mathrm e}^{3 x} c_1^{2} x^{2} \left (i \sqrt {3}-1\right )\right ) 2^{{1}/{3}}}{2 {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= \frac {\left (\frac {{\mathrm e}^{-3 x} 2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}} \left (i \sqrt {3}-1\right )}{2}+{\mathrm e}^{3 x} c_1^{2} x^{2} \left (1+i \sqrt {3}\right )\right ) 2^{{1}/{3}}}{2 {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 60.309 (sec). Leaf size: 383
ode=2*x*y[x]+3*x^2*y[x]+y[x]^3+(x^2+y[x]^2)*D[y[x],x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {e^{-3 x} \left (-2 e^{6 x} x^2+\sqrt [3]{2} \left (\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) e^{-3 x} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}}{2 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) e^{3 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) e^{3 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) e^{-3 x} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}}{2 \sqrt [3]{2}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*y(x) + 2*x*y(x) + (x**2 + y(x)**2)*Derivative(y(x), x) + y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out