83.22.12 problem 12
Internal
problem
ID
[19195]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
12
Date
solved
:
Monday, March 31, 2025 at 06:53:36 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} a^{2} y {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.192 (sec). Leaf size: 114
ode:=a^2*y(x)*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {a^{2} \textit {\_a}^{2}+\sqrt {-a^{2} \textit {\_a}^{2}+4}-2}{\textit {\_a} \left (a^{2} \textit {\_a}^{2}-3\right )}d \textit {\_a} +c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {a^{2} \textit {\_a}^{2}-\sqrt {-a^{2} \textit {\_a}^{2}+4}-2}{\textit {\_a} \left (a^{2} \textit {\_a}^{2}-3\right )}d \textit {\_a} +c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 60.886 (sec). Leaf size: 1287
ode=a^2*y[x]*D[y[x],x]^2-4*x*D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 16.252 (sec). Leaf size: 117
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a**2*y(x)*Derivative(y(x), x)**2 - 4*x*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} - \sqrt {3} \sqrt {\frac {x^{2}}{a^{2}}} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {3} \sqrt {\frac {x^{2}}{a^{2}}} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - x \sqrt {\frac {C_{1}}{x^{2}}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} x \sqrt {\frac {C_{1}}{x^{2}}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - \frac {2 x \sqrt {C_{1} + \log {\left (x^{2} \right )}}}{a} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {2 x \sqrt {C_{1} + \log {\left (x^{2} \right )}}}{a} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]