34.5.5 problem 21
Internal
problem
ID
[7957]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
9.
Equations
of
first
order
and
higher
degree.
Supplemetary
problems.
Page
65
Problem
number
:
21
Date
solved
:
Tuesday, September 30, 2025 at 05:12:13 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} 8 y {y^{\prime }}^{2}-2 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.052 (sec). Leaf size: 95
ode:=8*y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {2}\, x}{4} \\
y &= \frac {\sqrt {2}\, x}{4} \\
y &= 0 \\
\ln \left (x \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {8 y^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_1 &= 0 \\
\ln \left (x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {8 y^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.217 (sec). Leaf size: 174
ode=8*y[x]*D[y[x],x]^2-2*x*D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {e^{4 c_1} \sqrt {e^{8 c_1}-2 i x}}{2 \sqrt {2}}\\ y(x)&\to \frac {e^{4 c_1} \sqrt {e^{8 c_1}-2 i x}}{2 \sqrt {2}}\\ y(x)&\to -\frac {e^{4 c_1} \sqrt {2 i x+e^{8 c_1}}}{2 \sqrt {2}}\\ y(x)&\to \frac {e^{4 c_1} \sqrt {2 i x+e^{8 c_1}}}{2 \sqrt {2}}\\ y(x)&\to 0\\ y(x)&\to -\frac {x}{2 \sqrt {2}}\\ y(x)&\to \frac {x}{2 \sqrt {2}} \end{align*}
✓ Sympy. Time used: 135.995 (sec). Leaf size: 388
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*Derivative(y(x), x) + 8*y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}\right ]
\]