29.32.3 problem 937
Internal
problem
ID
[5515]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
937
Date
solved
:
Sunday, March 30, 2025 at 08:29:07 AM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} 3 x^{4} {y^{\prime }}^{2}-x y-y&=0 \end{align*}
✓ Maple. Time used: 0.459 (sec). Leaf size: 201
ode:=3*x^4*diff(y(x),x)^2-x*y(x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}+3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\
y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}-3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\
y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}-3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\
y &= \frac {\left (\operatorname {arctanh}\left (\sqrt {x +1}\right ) \sqrt {3}\, x \sqrt {x +1}+3 c_1 x \sqrt {x +1}+\sqrt {3}\, x +\sqrt {3}\right )^{2}}{36 \left (x +1\right ) x^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.143 (sec). Leaf size: 171
ode=3 x^4 (D[y[x],x])^2-x y[x]-y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {x^2 \text {arctanh}\left (\sqrt {x+1}\right )^2+2 x \text {arctanh}\left (\sqrt {x+1}\right ) \left (\sqrt {x+1}-\sqrt {3} c_1 x\right )+3 c_1{}^2 x^2+x-2 \sqrt {3} c_1 x \sqrt {x+1}+1}{12 x^2} \\
y(x)\to \frac {x^2 \text {arctanh}\left (\sqrt {x+1}\right )^2+2 x \text {arctanh}\left (\sqrt {x+1}\right ) \left (\sqrt {x+1}+\sqrt {3} c_1 x\right )+3 c_1{}^2 x^2+x+2 \sqrt {3} c_1 x \sqrt {x+1}+1}{12 x^2} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**4*Derivative(y(x), x)**2 - x*y(x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ZeroDivisionError : polynomial division