83.10.10 problem 10
Internal
problem
ID
[21979]
Book
:
Differential
Equations
By
Kaj
L.
Nielsen.
Second
edition
1966.
Barnes
and
nobel.
66-28306
Section
:
Chapter
IV.
First
order
differential
equations
of
higher
degree.
Ex.
XI
at
page
69
Problem
number
:
10
Date
solved
:
Thursday, October 02, 2025 at 08:21:19 PM
CAS
classification
:
[_quadrature]
\begin{align*} 2 {y^{\prime }}^{2}+y y^{\prime }-y^{4}&=0 \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 97
ode:=2*diff(y(x),x)^2+y(x)*diff(y(x),x)-y(x)^4 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
\frac {\left (-8 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1+8 y^{2}}}\right )+4 x -4 c_1 \right ) y^{2}-\sqrt {1+8 y^{2}}+1}{4 y^{2}} &= 0 \\
\frac {\left (8 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1+8 y^{2}}}\right )+4 x -4 c_1 \right ) y^{2}+\sqrt {1+8 y^{2}}+1}{4 y^{2}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.736 (sec). Leaf size: 136
ode=2*D[y[x],x]^2+y[x]*D[y[x],x]-y[x]^4==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {1}{2} \text {arctanh}\left (\frac {\text {$\#$1}}{\sqrt {8 \text {$\#$1}^4+\text {$\#$1}^2}}\right )-\frac {\text {$\#$1}}{2 \left (\sqrt {8 \text {$\#$1}^4+\text {$\#$1}^2}-\text {$\#$1}\right )}\&\right ]\left [\frac {x}{4}+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [-\frac {1}{2} \text {arctanh}\left (\frac {\text {$\#$1}}{\sqrt {8 \text {$\#$1}^4+\text {$\#$1}^2}}\right )-\frac {\text {$\#$1}}{2 \left (\sqrt {8 \text {$\#$1}^4+\text {$\#$1}^2}+\text {$\#$1}\right )}\&\right ]\left [-\frac {x}{4}+c_1\right ]\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 6.518 (sec). Leaf size: 230
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)**4 + y(x)*Derivative(y(x), x) + 2*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ x - 2 \operatorname {asinh}{\left (\frac {\sqrt {2}}{4 y{\left (x \right )}} \right )} + \frac {1}{4 y^{2}{\left (x \right )}} - \frac {\sqrt {2}}{2 \sqrt {1 + \frac {1}{8 y^{2}{\left (x \right )}}} y{\left (x \right )}} - \frac {\sqrt {2}}{16 \sqrt {1 + \frac {1}{8 y^{2}{\left (x \right )}}} y^{3}{\left (x \right )}} = C_{1}, \ - 4 \left (\begin {cases} \frac {\operatorname {asinh}{\left (\frac {\sqrt {2}}{4 y{\left (x \right )}} \right )}}{2} - \frac {1}{16 y^{2}{\left (x \right )}} + \frac {\sqrt {2}}{8 \sqrt {1 + \frac {1}{8 y^{2}{\left (x \right )}}} y{\left (x \right )}} + \frac {\sqrt {2}}{64 \sqrt {1 + \frac {1}{8 y^{2}{\left (x \right )}}} y^{3}{\left (x \right )}} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 8 \\- \frac {\operatorname {asinh}{\left (\frac {\sqrt {2}}{4 y{\left (x \right )}} \right )}}{2} - \frac {1}{16 y^{2}{\left (x \right )}} - \frac {\sqrt {2}}{8 \sqrt {1 + \frac {1}{8 y^{2}{\left (x \right )}}} y{\left (x \right )}} - \frac {\sqrt {2}}{64 \sqrt {1 + \frac {1}{8 y^{2}{\left (x \right )}}} y^{3}{\left (x \right )}} & \text {otherwise} \end {cases}\right ) = C_{1} - x\right ]
\]