60.1.282 problem 288
Internal
problem
ID
[10296]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
288
Date
solved
:
Sunday, March 30, 2025 at 03:44:43 PM
CAS
classification
:
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\begin{align*} \left (6 y^{2}-3 x^{2} y+1\right ) y^{\prime }-3 x y^{2}+x&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 583
ode:=(6*y(x)^2-3*x^2*y(x)+1)*diff(y(x),x)-3*x*y(x)^2+x = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{1}/{3}}}{12}+\frac {3 x^{4}-8}{4 \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{1}/{3}}}+\frac {x^{2}}{4} \\
y &= \frac {9 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{2}/{3}}-9 x^{4}+6 x^{2} \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{1}/{3}}-24 i \sqrt {3}-\left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{2}/{3}}+24}{24 \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{1}/{3}}} \\
y &= -\frac {9 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{2}/{3}}+9 x^{4}-6 x^{2} \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{1}/{3}}-24 i \sqrt {3}+\left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{2}/{3}}-24}{24 \left (-324 x^{2}-432 c_1 +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_1 \,x^{6}+621 x^{4}+1944 c_1 \,x^{2}+1296 c_1^{2}+96}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.807 (sec). Leaf size: 538
ode=(6*y[x]^2-3*x^2*y[x]+1)*D[y[x],x]-3*x*y[x]^2+x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{36} \left (9 x^2-3 \sqrt [3]{3} \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}-\frac {3\ 3^{2/3} \left (3 x^4-8\right )}{\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right ) \\
y(x)\to \frac {1}{24} \left (6 x^2+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}+\frac {3^{2/3} \left (1+i \sqrt {3}\right ) \left (3 x^4-8\right )}{\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right ) \\
y(x)\to \frac {1}{24} \left (6 x^2+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}+\frac {3^{2/3} \left (1-i \sqrt {3}\right ) \left (3 x^4-8\right )}{\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right ) \\
y(x)\to -\frac {1}{\sqrt {3}} \\
y(x)\to \frac {1}{\sqrt {3}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*x*y(x)**2 + x + (-3*x**2*y(x) + 6*y(x)**2 + 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out