71.8.5 problem 4 (a)
Internal
problem
ID
[14368]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
2.
The
Initial
Value
Problem.
Exercises
2.4.4,
page
115
Problem
number
:
4
(a)
Date
solved
:
Monday, March 31, 2025 at 12:19:40 PM
CAS
classification
:
[_linear]
\begin{align*} y^{\prime }&=\frac {y}{-x^{2}+1}+\sqrt {x} \end{align*}
With initial conditions
\begin{align*} y \left (\frac {1}{2}\right )&=1 \end{align*}
✓ Maple. Time used: 0.453 (sec). Leaf size: 142
ode:=diff(y(x),x) = y(x)/(-x^2+1)+x^(1/2);
ic:=y(1/2) = 1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {\left (x +1\right ) \left (12 i \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )-8 i \operatorname {EllipticF}\left (\frac {\sqrt {3}}{2}, \sqrt {2}\right )-\sqrt {3}\, \sqrt {2}+2 \sqrt {3}\right )}{6 \sqrt {-x^{2}+1}}+\frac {-2 \sqrt {x +1}\, \sqrt {-2 x +2}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {x +1}\, \sqrt {-2 x +2}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+2 x^{3}-2 x}{\sqrt {x}\, \left (3 x -3\right )}
\]
✓ Mathematica. Time used: 0.123 (sec). Leaf size: 69
ode=D[y[x],x]==y[x]/(1-x^2)+Sqrt[x];
ic={y[1/2]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \exp \left (\int _{\frac {1}{2}}^x\frac {1}{1-K[1]^2}dK[1]\right ) \left (\int _{\frac {1}{2}}^x\exp \left (-\int _{\frac {1}{2}}^{K[2]}\frac {1}{1-K[1]^2}dK[1]\right ) \sqrt {K[2]}dK[2]+1\right )
\]
✓ Sympy. Time used: 57.578 (sec). Leaf size: 345
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sqrt(x) + Derivative(y(x), x) - y(x)/(1 - x**2),0)
ics = {y(1/2): 1}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \begin {cases} \frac {\sqrt {x + 1} \left (\int \limits ^{\frac {1}{2}} \frac {\sqrt {x}}{x \sqrt {x - 1} \sqrt {x + 1} + \sqrt {x - 1} \sqrt {x + 1}}\, dx - \int \limits ^{\frac {1}{2}} \frac {x^{\frac {5}{2}}}{x \sqrt {x - 1} \sqrt {x + 1} + \sqrt {x - 1} \sqrt {x + 1}}\, dx + \frac {\sqrt {3} i}{3}\right )}{- i \sqrt {1 - x} + \sqrt {x - 1} + \sqrt {x + 1} \left (\begin {cases} \frac {\sqrt {x - 1}}{\sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \sqrt {1 - x}}{\sqrt {x + 1}} & \text {otherwise} \end {cases}\right )} - \frac {\sqrt {x + 1} \int \frac {\sqrt {x}}{x \sqrt {x - 1} \sqrt {x + 1} + \sqrt {x - 1} \sqrt {x + 1}}\, dx}{- i \sqrt {1 - x} + \sqrt {x - 1} + \sqrt {x + 1} \left (\begin {cases} \frac {\sqrt {x - 1}}{\sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \sqrt {1 - x}}{\sqrt {x + 1}} & \text {otherwise} \end {cases}\right )} + \frac {\sqrt {x + 1} \int \frac {x^{\frac {5}{2}}}{x \sqrt {x - 1} \sqrt {x + 1} + \sqrt {x - 1} \sqrt {x + 1}}\, dx}{- i \sqrt {1 - x} + \sqrt {x - 1} + \sqrt {x + 1} \left (\begin {cases} \frac {\sqrt {x - 1}}{\sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \sqrt {1 - x}}{\sqrt {x + 1}} & \text {otherwise} \end {cases}\right )} & \text {for}\: x \geq -3 \wedge x \leq 1 \\\text {NaN} & \text {otherwise} \end {cases}
\]