8.5.33 problem 33
Internal
problem
ID
[761]
Book
:
Differential
equations
and
linear
algebra,
3rd
ed.,
Edwards
and
Penney
Section
:
Section
1.6,
Substitution
methods
and
exact
equations.
Page
74
Problem
number
:
33
Date
solved
:
Saturday, March 29, 2025 at 10:20:15 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 3 x^{2}+2 y^{2}+\left (4 x y+6 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 401
ode:=3*x^2+2*y(x)^2+(4*x*y(x)+6*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\frac {\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}}{2}+\frac {2 x^{2} c_1^{2}}{\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}}-c_1 x}{3 c_1} \\
y &= \frac {4 i \sqrt {3}\, c_1^{2} x^{2}-i \sqrt {3}\, \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{2}/{3}}-4 x^{2} c_1^{2}-4 c_1 x \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}-\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{2}/{3}}}{12 \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{2}/{3}}-4 x c_1 \left (i x c_1 \sqrt {3}+c_1 x +\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}\right )}{12 \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 x^{6} c_1^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 40.356 (sec). Leaf size: 679
ode=3*x^2+2*y[x]^2+(4*x*y[x]+6*y[x]^2)*D[y[x],x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-124 x^3+\sqrt {-256 x^6+\left (-124 x^3+108 e^{2 c_1}\right ){}^2}+108 e^{2 c_1}}}{6 \sqrt [3]{2}}+\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{-124 x^3+\sqrt {-256 x^6+\left (-124 x^3+108 e^{2 c_1}\right ){}^2}+108 e^{2 c_1}}}-\frac {x}{3} \\
y(x)\to \frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}-\frac {i \left (\sqrt {3}-i\right ) x^2}{3 \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}}-\frac {x}{3} \\
y(x)\to -\frac {1}{12} i \left (\sqrt {3}-i\right ) \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{3 \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}}-\frac {x}{3} \\
y(x)\to \frac {1}{6} \left (\sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}+\frac {2\ 2^{2/3} x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-2 x\right ) \\
y(x)\to \frac {1}{12} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}-\frac {2 i 2^{2/3} \left (\sqrt {3}-i\right ) x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-4 x\right ) \\
y(x)\to \frac {1}{12} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}+\frac {2 i 2^{2/3} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-4 x\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2 + (4*x*y(x) + 6*y(x)**2)*Derivative(y(x), x) + 2*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out