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
:
Monday, January 27, 2025 at 11:34:54 AM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} 3 x^{4} {y^{\prime }}^{2}-y x -y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 201
dsolve(3*x^4*diff(y(x),x)^2-x*y(x)-y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \frac {\left (\sqrt {3}\, \operatorname {arctanh}\left (\sqrt {x +1}\right ) 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 \left (x \right ) &= \frac {\left (\sqrt {3}\, \operatorname {arctanh}\left (\sqrt {x +1}\right ) 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 \left (x \right ) &= \frac {\left (\sqrt {3}\, \operatorname {arctanh}\left (\sqrt {x +1}\right ) 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 \left (x \right ) &= \frac {\left (\sqrt {3}\, \operatorname {arctanh}\left (\sqrt {x +1}\right ) 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*}
✓ Solution by Mathematica
Time used: 0.143 (sec). Leaf size: 171
DSolve[3 x^4 (D[y[x],x])^2-x y[x]-y[x]==0,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*}