82.12.9 problem Ex. 9

Internal problem ID [18695]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 9
Date solved : Thursday, March 13, 2025 at 12:38:57 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }+\frac {y}{\sqrt {-x^{2}+1}}&=\frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 39

ode:=diff(y(x),x)+y(x)/(-x^2+1)^(1/2) = (x+(-x^2+1)^(1/2))/(-x^2+1)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \left (\int \frac {{\mathrm e}^{\arcsin \left (x \right )} \left (x +\sqrt {-x^{2}+1}\right )}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x +c_{1} \right ) {\mathrm e}^{-\arcsin \left (x \right )} \]

✓ Mathematica. Time used: 5.553 (sec). Leaf size: 164

ode=D[y[x],x]+y[x]/Sqrt[1-x^2]==(x+Sqrt[1-x^2])/(1-x^2)^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (\frac {1}{20}+\frac {i}{40}\right ) \left (\frac {5 i \left (1+e^{2 i \arcsin (x)}\right )^2 \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (x)}\right )}{x^2-1}+(1+2 i) \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},-e^{2 i \arcsin (x)}\right )-e^{2 i \arcsin (x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2},2-\frac {i}{2},-e^{2 i \arcsin (x)}\right )+(16-8 i) c_1 e^{-\arcsin (x)}+\frac {(2-i) \left (x^3+4 \sqrt {1-x^2}-x\right )}{\left (1-x^2\right )^{3/2}}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x + sqrt(1 - x**2))/(1 - x**2)**2 + y(x)/sqrt(1 - x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out