29.34.10 problem 1012
Internal
problem
ID
[5581]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1012
Date
solved
:
Monday, January 27, 2025 at 12:11:35 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 3 x y^{4} {y^{\prime }}^{2}-y^{5} y^{\prime }+1&=0 \end{align*}
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 289
dsolve(3*x*y(x)^4*diff(y(x),x)^2-y(x)^5*diff(y(x),x)+1 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 3^{{1}/{6}} 2^{{1}/{3}} x^{{1}/{6}} \\
y \left (x \right ) &= -3^{{1}/{6}} 2^{{1}/{3}} x^{{1}/{6}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{{1}/{6}} 2^{{1}/{3}} x^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 3^{{1}/{6}} 2^{{1}/{3}} x^{{1}/{6}}}{2} \\
y \left (x \right ) &= -\frac {\left (i \sqrt {3}-1\right ) 3^{{1}/{6}} 2^{{1}/{3}} x^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 3^{{1}/{6}} 2^{{1}/{3}} x^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {3^{{1}/{6}} \left (-\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{c_{1}} \\
y \left (x \right ) &= -\frac {3^{{1}/{6}} \left (-\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{c_{1}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{{1}/{6}} \left (-\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {\left (i 3^{{2}/{3}}-3^{{1}/{6}}\right ) \left (-\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y \left (x \right ) &= -\frac {\left (i \sqrt {3}-1\right ) 3^{{1}/{6}} \left (-\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {\left (i 3^{{2}/{3}}+3^{{1}/{6}}\right ) \left (-\left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.056 (sec). Leaf size: 230
DSolve[3 x y[x]^4 (D[y[x],x])^2 -y[x]^5 D[y[x],x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt [3]{-\frac {1}{2}} e^{-\frac {c_1}{6}} \sqrt [3]{12 x+e^{c_1}} \\
y(x)\to e^{-\frac {c_1}{6}} \sqrt [3]{6 x+\frac {e^{c_1}}{2}} \\
y(x)\to (-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{6 x+\frac {e^{c_1}}{2}} \\
y(x)\to -\sqrt [3]{-2} \sqrt [6]{3} \sqrt [6]{x} \\
y(x)\to \sqrt [3]{-2} \sqrt [6]{3} \sqrt [6]{x} \\
y(x)\to -\sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\
y(x)\to \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\
y(x)\to -(-1)^{2/3} \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\
y(x)\to (-1)^{2/3} \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\
\end{align*}