10.6.21 problem 21
Internal
problem
ID
[1238]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Miscellaneous
problems,
end
of
chapter
2.
Page
133
Problem
number
:
21
Date
solved
:
Saturday, March 29, 2025 at 10:49:32 PM
CAS
classification
:
[_rational]
\begin{align*} \frac {-4+6 x y+2 y^{2}}{3 x^{2}+4 x y+3 y^{2}}+y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 513
ode:=(-4+6*x*y(x)+2*y(x)^2)/(3*x^2+4*x*y(x)+3*y(x)^2)+diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}}{6}-\frac {10 x^{2}}{3 \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}}-\frac {2 x}{3} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}}{12}-\frac {5 \left (i \sqrt {3}\, x -x +\frac {2 \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}}{5}\right ) x}{3 \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{2}/{3}}+20 i \sqrt {3}\, x^{2}-\left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{2}/{3}}-8 x \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}+20 x^{2}}{12 \left (152 x^{3}-108 c_1 +432 x +12 \sqrt {216 x^{6}-228 c_1 \,x^{3}+912 x^{4}+81 c_1^{2}-648 c_1 x +1296 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.888 (sec). Leaf size: 383
ode=(-4+6*x*y[x]+2*y[x]^2)/(3*x^2+4*x*y[x]+3*y[x]^2)+D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}-\frac {10 \sqrt [3]{2} x^2}{\sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}}-4 x\right ) \\
y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}+\frac {10 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}}-8 x\right ) \\
y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}+\frac {10 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}}-8 x\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) + (6*x*y(x) + 2*y(x)**2 - 4)/(3*x**2 + 4*x*y(x) + 3*y(x)**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out