29.34.11 problem 1013
Internal
problem
ID
[5582]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1013
Date
solved
:
Monday, January 27, 2025 at 12:12:26 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-a&=0 \end{align*}
✓ Solution by Maple
Time used: 0.089 (sec). Leaf size: 271
dsolve(9*x*y(x)^4*diff(y(x),x)^2-3*y(x)^5*diff(y(x),x)-a = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}} \\
y \left (x \right ) &= -2^{{1}/{3}} \left (-a x \right )^{{1}/{6}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-a x \right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (a \left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{c_{1}} \\
y \left (x \right ) &= -\frac {\left (a \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 ) \left (a \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 ) \left (a \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 ) \left (a \left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) \left (a \left (-x +c_{1} \right )^{2} c_{1}^{5}\right )^{{1}/{6}}}{2 c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 9.960 (sec). Leaf size: 358
DSolve[9 x y[x]^4 (D[y[x],x])^2 -3 y[x]^5 D[y[x],x]-a==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt [3]{-\frac {1}{2}} e^{-\frac {c_1}{6}} \sqrt [3]{-4 a x+e^{c_1}} \\
y(x)\to \frac {e^{-\frac {c_1}{6}} \sqrt [3]{-4 a x+e^{c_1}}}{\sqrt [3]{2}} \\
y(x)\to \frac {(-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{-4 a x+e^{c_1}}}{\sqrt [3]{2}} \\
y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 a x+e^{c_1}\right )} \\
y(x)\to \frac {\sqrt [3]{e^{-\frac {c_1}{2}} \left (4 a x-e^{c_1}\right )}}{\sqrt [3]{2}} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (-4 a x+e^{c_1}\right )}}{\sqrt [3]{2}} \\
y(x)\to -i \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \\
y(x)\to i \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \\
y(x)\to -\sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \\
y(x)\to \sqrt [6]{-1} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \\
y(x)\to -(-1)^{5/6} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \\
y(x)\to (-1)^{5/6} \sqrt [3]{2} \sqrt [6]{a} \sqrt [6]{x} \\
\end{align*}