87.6.23 problem 26
Internal
problem
ID
[23340]
Book
:
Ordinary
differential
equations
with
modern
applications.
Ladas,
G.
E.
and
Finizio,
N.
Wadsworth
Publishing.
California.
1978.
ISBN
0-534-00552-7.
QA372.F56
Section
:
Chapter
1.
Elementary
methods.
First
order
differential
equations.
Exercise
at
page
53
Problem
number
:
26
Date
solved
:
Thursday, October 02, 2025 at 09:38:50 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} y^{\prime }&=\frac {\sqrt {2}\, \sqrt {\frac {x +y}{x}}}{2} \end{align*}
With initial conditions
\begin{align*}
y \left (1\right )&=2 \\
\end{align*}
✓ Maple. Time used: 0.762 (sec). Leaf size: 90
ode:=diff(y(x),x) = 1/2*2^(1/2)*((x+y(x))/x)^(1/2);
ic:=[y(1) = 2];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = \operatorname {RootOf}\left (4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {\frac {x +\textit {\_Z}}{x}}}{2}\right )-2 \,\operatorname {arctanh}\left (\sqrt {2}\, \sqrt {\frac {x +\textit {\_Z}}{x}}\right )-\ln \left (\frac {2 \textit {\_Z} +x}{x}\right )-2 \ln \left (\frac {\textit {\_Z} -x}{x}\right )-3 \ln \left (x \right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {3}}{2}\right )+2 \,\operatorname {arctanh}\left (\sqrt {2}\, \sqrt {3}\right )+\ln \left (5\right )\right )
\]
✓ Mathematica. Time used: 11.833 (sec). Leaf size: 356
ode=D[y[x],x]==Sqrt[ (x+y[x])/(2*x) ];
ic={y[1]==2};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {Root}\left [\frac {4 \text {$\#$1}^6}{\left (103+42 \sqrt {6}\right )^2}-\frac {12 \text {$\#$1}^5 x}{\left (103+42 \sqrt {6}\right )^2}+\frac {9 \text {$\#$1}^4 x^2}{\left (103+42 \sqrt {6}\right )^2}+\text {$\#$1}^3 \left (\frac {4 x^3}{\left (103+42 \sqrt {6}\right )^2}-\frac {4}{103+42 \sqrt {6}}\right )+\text {$\#$1}^2 \left (-\frac {6 x^4}{\left (103+42 \sqrt {6}\right )^2}-\frac {30 x}{103+42 \sqrt {6}}\right )-\frac {24 \text {$\#$1} x^2}{103+42 \sqrt {6}}+\frac {x^6}{\left (103+42 \sqrt {6}\right )^2}-\frac {6 x^3}{103+42 \sqrt {6}}+1\&,1\right ]\\ y(x)&\to \text {Root}\left [\left (84772+34608 \sqrt {6}\right ) \text {$\#$1}^6+\text {$\#$1}^5 \left (-103824 \sqrt {6} x-254316 x\right )+\text {$\#$1}^4 \left (77868 \sqrt {6} x^2+190737 x^2\right )+\text {$\#$1}^3 \left (34608 \sqrt {6} x^3+84772 x^3-4200 \sqrt {6}-10300\right )+\text {$\#$1}^2 \left (-51912 \sqrt {6} x^4-127158 x^4-31500 \sqrt {6} x-77250 x\right )+\text {$\#$1} \left (-25200 \sqrt {6} x^2-61800 x^2\right )+8652 \sqrt {6} x^6+21193 x^6-6300 \sqrt {6} x^3-15450 x^3+625\&,4\right ] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sqrt(2)*sqrt((x + y(x))/x)/2 + Derivative(y(x), x),0)
ics = {y(1): 2}
dsolve(ode,func=y(x),ics=ics)
ValueError : Couldnt solve for initial conditions