82.14.3 problem Ex. 3
Internal
problem
ID
[18738]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
III.
Equations
of
the
first
order
but
not
of
the
first
degree.
Problems
at
page
33
Problem
number
:
Ex.
3
Date
solved
:
Monday, March 31, 2025 at 06:07:14 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} 4 y&=x^{2}+{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.557 (sec). Leaf size: 136
ode:=4*y(x) = x^2+diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{\frac {c_1}{2}}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{\frac {c_1}{2}}}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{\frac {c_1}{2}}}{2}\right )^{2}} \\
y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_1}{2}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_1}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_1}{2}\right )^{2}} \\
y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_1}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_1}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_1}{2}\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 2.58 (sec). Leaf size: 162
ode=4*y[x]==x^2+D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )+\frac {x \left (-\sqrt {4 y(x)-x^2}\right )+\left (x^2-2 y(x)\right ) \log \left (2 y(x)-x^2\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {x \sqrt {4 y(x)-x^2}+\left (x^2-2 y(x)\right ) \log \left (2 y(x)-x^2\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}-\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + 4*y(x) - Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-x**2 + 4*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method