29.10.11 problem 277
Internal
problem
ID
[4877]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
10
Problem
number
:
277
Date
solved
:
Sunday, March 30, 2025 at 04:08:37 AM
CAS
classification
:
[_linear]
\begin{align*} \left (-x^{2}+1\right ) y^{\prime }&=1-x^{2}+y \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 30
ode:=(-x^2+1)*diff(y(x),x) = 1-x^2+y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\sqrt {-x^{2}+1}+\arcsin \left (x \right )+c_1 \right ) \left (x +1\right )}{\sqrt {-x^{2}+1}}
\]
✓ Mathematica. Time used: 0.106 (sec). Leaf size: 56
ode=(1-x^2)D[y[x],x]==1-x^2+y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {\sqrt {x+1} \left (-2 \arctan \left (\frac {\sqrt {1-x^2}}{x-1}\right )+\sqrt {1-x^2}+c_1\right )}{\sqrt {1-x}}
\]
✓ Sympy. Time used: 12.488 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + (1 - x**2)*Derivative(y(x), x) - y(x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \begin {cases} \frac {C_{1} \sqrt {x + 1}}{- i \sqrt {1 - x} + \sqrt {x - 1} + \sqrt {x + 1} \left (\begin {cases} \frac {\sqrt {x - 1}}{\sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \sqrt {1 - x}}{\sqrt {x + 1}} & \text {otherwise} \end {cases}\right )} - \frac {i \sqrt {1 - x}}{- i \sqrt {1 - x} + \sqrt {x - 1} + \sqrt {x + 1} \left (\begin {cases} \frac {\sqrt {x - 1}}{\sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \sqrt {1 - x}}{\sqrt {x + 1}} & \text {otherwise} \end {cases}\right )} + \frac {\sqrt {x + 1} \int \frac {x^{2}}{x \sqrt {x - 1} \sqrt {x + 1} + \sqrt {x - 1} \sqrt {x + 1}}\, dx}{- i \sqrt {1 - x} + \sqrt {x - 1} + \sqrt {x + 1} \left (\begin {cases} \frac {\sqrt {x - 1}}{\sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \sqrt {1 - x}}{\sqrt {x + 1}} & \text {otherwise} \end {cases}\right )} & \text {for}\: x \geq -3 \wedge x \leq 1 \\\text {NaN} & \text {otherwise} \end {cases}
\]