84.14.10 problem 7.18
Internal
problem
ID
[22167]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
7.
Integrating
factors.
Supplementary
problems
Problem
number
:
7.18
Date
solved
:
Thursday, October 02, 2025 at 08:33:12 PM
CAS
classification
:
[_separable]
\begin{align*} 2 x y^{2}+\frac {x}{y^{2}}+4 x^{2} y y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 103
ode:=2*x*y(x)^2+x/y(x)^2+4*x^2*y(x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {2^{{3}/{4}} {\left (-\left (x^{2}-2 c_1 \right ) x^{2}\right )}^{{1}/{4}}}{2 x} \\
y &= \frac {2^{{3}/{4}} {\left (-\left (x^{2}-2 c_1 \right ) x^{2}\right )}^{{1}/{4}}}{2 x} \\
y &= -\frac {i 2^{{3}/{4}} {\left (-\left (x^{2}-2 c_1 \right ) x^{2}\right )}^{{1}/{4}}}{2 x} \\
y &= \frac {i 2^{{3}/{4}} {\left (-\left (x^{2}-2 c_1 \right ) x^{2}\right )}^{{1}/{4}}}{2 x} \\
\end{align*}
✓ Mathematica. Time used: 0.184 (sec). Leaf size: 284
ode=(2*x*y[x]^2+x/y[x]^2)+(4*x^2*y[x])*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt [4]{-x^2+e^{8 c_1}}}{\sqrt [4]{2} \sqrt {x}}\\ y(x)&\to -\frac {i \sqrt [4]{-x^2+e^{8 c_1}}}{\sqrt [4]{2} \sqrt {x}}\\ y(x)&\to \frac {i \sqrt [4]{-x^2+e^{8 c_1}}}{\sqrt [4]{2} \sqrt {x}}\\ y(x)&\to \frac {\sqrt [4]{-x^2+e^{8 c_1}}}{\sqrt [4]{2} \sqrt {x}}\\ y(x)&\to -\sqrt [4]{-\frac {1}{2}}\\ y(x)&\to \sqrt [4]{-\frac {1}{2}}\\ y(x)&\to \frac {1-i}{2^{3/4}}\\ y(x)&\to -\frac {1-i}{2^{3/4}}\\ y(x)&\to \frac {i x^{3/2}}{\sqrt [4]{2} \left (-x^2\right )^{3/4}}\\ y(x)&\to \frac {x^{3/2}}{\sqrt [4]{2} \left (-x^2\right )^{3/4}}\\ y(x)&\to \frac {i \sqrt [4]{-x^2}}{\sqrt [4]{2} \sqrt {x}}\\ y(x)&\to \frac {\sqrt [4]{-x^2}}{\sqrt [4]{2} \sqrt {x}} \end{align*}
✓ Sympy. Time used: 1.394 (sec). Leaf size: 71
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**2*y(x)*Derivative(y(x), x) + 2*x*y(x)**2 + x/y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\left (-2\right )^{\frac {3}{4}} \sqrt [4]{\frac {C_{1}}{x^{2}} + 1}}{2}, \ y{\left (x \right )} = \frac {\left (-2\right )^{\frac {3}{4}} \sqrt [4]{\frac {C_{1}}{x^{2}} + 1}}{2}, \ y{\left (x \right )} = - \sqrt [4]{\frac {C_{1}}{x^{2}} - \frac {1}{2}}, \ y{\left (x \right )} = \sqrt [4]{\frac {C_{1}}{x^{2}} - \frac {1}{2}}\right ]
\]