89.31.5 problem 5
Internal
problem
ID
[24950]
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
:
5
Date
solved
:
Thursday, October 02, 2025 at 11:36:30 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} 4 y {y^{\prime }}^{2}-2 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.102 (sec). Leaf size: 73
ode:=4*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 {x}{2} \\
y &= \frac {x}{2} \\
y &= 0 \\
y &= \sqrt {c_1 \left (-i x +c_1 \right )} \\
y &= \sqrt {c_1 \left (i x +c_1 \right )} \\
y &= -\sqrt {c_1 \left (-i x +c_1 \right )} \\
y &= -\sqrt {c_1 \left (i x +c_1 \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.564 (sec). Leaf size: 84
ode=4*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 {1}{2} \sqrt {2 e^{4 c_1} x-e^{8 c_1}}\\ y(x)&\to \frac {1}{2} \sqrt {2 e^{4 c_1} x-e^{8 c_1}}\\ y(x)&\to 0\\ y(x)&\to -\frac {x}{2}\\ y(x)&\to \frac {x}{2} \end{align*}
✓ Sympy. Time used: 149.896 (sec). Leaf size: 469
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*Derivative(y(x), x) + 4*y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{4}\right ]
\]