29.34.7 problem 1003
Internal
problem
ID
[5578]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1003
Date
solved
:
Monday, January 27, 2025 at 12:11:08 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }-a&=0 \end{align*}
✓ Solution by Maple
Time used: 0.133 (sec). Leaf size: 155
dsolve(2*x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)-a = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\
y \left (x \right ) &= -2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\
y \left (x \right ) &= -i 2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\
y \left (x \right ) &= i 2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\
y \left (x \right ) &= \frac {2^{{1}/{4}} \left (a \left (-x +c_{1} \right )^{2} c_{1}^{3}\right )^{{1}/{4}}}{c_{1}} \\
y \left (x \right ) &= -\frac {2^{{1}/{4}} \left (a \left (-x +c_{1} \right )^{2} c_{1}^{3}\right )^{{1}/{4}}}{c_{1}} \\
y \left (x \right ) &= -\frac {i 2^{{1}/{4}} \left (a \left (-x +c_{1} \right )^{2} c_{1}^{3}\right )^{{1}/{4}}}{c_{1}} \\
y \left (x \right ) &= \frac {i 2^{{1}/{4}} \left (a \left (-x +c_{1} \right )^{2} c_{1}^{3}\right )^{{1}/{4}}}{c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.527 (sec). Leaf size: 151
DSolve[2 x y[x]^2 (D[y[x],x])^2 -y[x]^3 D[y[x],x] -a ==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {e^{-\frac {c_1}{4}} \sqrt {-8 a x+e^{c_1}}}{\sqrt {2}} \\
y(x)\to \frac {e^{-\frac {c_1}{4}} \sqrt {-8 a x+e^{c_1}}}{\sqrt {2}} \\
y(x)\to -(-2)^{3/4} \sqrt [4]{a} \sqrt [4]{x} \\
y(x)\to (-2)^{3/4} \sqrt [4]{a} \sqrt [4]{x} \\
y(x)\to (-1-i) \sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x} \\
y(x)\to (1+i) \sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x} \\
\end{align*}