23.2.78 problem 80
Internal
problem
ID
[5433]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
80
Date
solved
:
Tuesday, September 30, 2025 at 12:42:19 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: 0.196 (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: 0.395 (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: 73.924 (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)
\[
\left [ y{\left (x \right )} = \begin {cases} - \frac {1}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} & \text {for}\: \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} > -1 \wedge \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {1}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} & \text {for}\: \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} > -1 \wedge \frac {x}{\sqrt {2 x e^{- C_{1}} - e^{- 2 C_{1}}}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - C_{1} \sqrt {\frac {1}{2 C_{1} x - 1}} & \text {for}\: C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} > -1 \wedge C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} \sqrt {\frac {1}{2 C_{1} x - 1}} & \text {for}\: C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} > -1 \wedge C_{1} x \sqrt {\frac {1}{2 C_{1} x - 1}} < 1 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]