89.31.8 problem 8
Internal
problem
ID
[24953]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
16.
Equations
of
order
one
and
higher
degree.
Exercises
at
page
246
Problem
number
:
8
Date
solved
:
Thursday, October 02, 2025 at 11:36:32 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{2}+2 x y^{3} y^{\prime }+y^{4}&=0 \end{align*}
✓ Maple. Time used: 0.197 (sec). Leaf size: 48
ode:=diff(y(x),x)^2+2*x*y(x)^3*diff(y(x),x)+y(x)^4 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {1}{x} \\
y &= \frac {1}{x} \\
y &= 0 \\
y &= \frac {1}{\sqrt {-c_1 \left (c_1 -2 x \right )}} \\
y &= -\frac {1}{\sqrt {c_1 \left (-c_1 +2 x \right )}} \\
\end{align*}
✓ Mathematica. Time used: 0.398 (sec). Leaf size: 171
ode=D[y[x],x]^2+2*x*y[x]^3*D[y[x],x]+y[x]^4==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solve}\left [-\frac {\sqrt {1-x^2 y(x)^2} y(x)^3 \text {arcsinh}\left (x \sqrt {-y(x)^2}\right )}{\sqrt {-y(x)^2} \sqrt {y(x)^4 \left (x^2 y(x)^2-1\right )}}-\log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {y(x)^3 \sqrt {1-x^2 y(x)^2} \text {arcsinh}\left (x \sqrt {-y(x)^2}\right )}{\sqrt {-y(x)^2} \sqrt {y(x)^4 \left (x^2 y(x)^2-1\right )}}-\log (y(x))=c_1,y(x)\right ]\\ y(x)&\to 0\\ y(x)&\to -\frac {1}{x}\\ y(x)&\to \frac {1}{x} \end{align*}
✓ Sympy. Time used: 73.329 (sec). Leaf size: 218
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x)**3*Derivative(y(x), x) + y(x)**4 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} - \frac {1}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} & \text {for}\: \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} > -1 \wedge \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {1}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} & \text {for}\: \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} > -1 \wedge \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - C_{1} \sqrt {\frac {1}{2 C_{1} x - 1}} & \text {for}\: C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} > -1 \wedge C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} \sqrt {\frac {1}{2 C_{1} x - 1}} & \text {for}\: C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} > -1 \wedge C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]