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
:
Tuesday, March 04, 2025 at 10:28:28 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }-a&=0 \end{align*}
✓ Maple. Time used: 0.133 (sec). Leaf size: 155
ode:=2*x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)-a = 0;
dsolve(ode,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*}
✓ Mathematica. Time used: 1.527 (sec). Leaf size: 151
ode=2 x y[x]^2 (D[y[x],x])^2 -y[x]^3 D[y[x],x] -a ==0;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + 2*x*y(x)**2*Derivative(y(x), x)**2 - y(x)**3*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out