82.54.9 problem Ex. 9

Internal problem ID [18954]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 9
Date solved : Monday, March 31, 2025 at 06:26:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y&=x \left (-x^{2}+1\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 49

ode:=(-x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = x*(-x^2+1)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sqrt {-x^{2}+1}\, \left (x^{3}-x \right )}{9}+\left (\ln \left (x +\sqrt {x^{2}-1}\right ) c_1 +c_2 \right ) x -\sqrt {x^{2}-1}\, c_1 \]

✓ Mathematica. Time used: 0.352 (sec). Leaf size: 173

ode=(1-x^2)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==x*(1-x^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\left (x^2-1\right ) \left (x^5-2 x^3-9 c_1 \sqrt {1-x^2} x+9 c_2 \sqrt {-\left (x^2-1\right )^2}+x\right )-3 x \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right ) \left (3 c_2 \sqrt {1-x^2} x^2-3 c_2 \sqrt {1-x^2}+\sqrt {x^2-1} x^5-\left (\sqrt {x^2-1}-\sqrt {1-x^2} \sqrt {-\left (x^2-1\right )^2}\right ) x^3\right )}{9 \left (1-x^2\right )^{3/2}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(1 - x**2)**(3/2) + x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(-x**2*sqrt(1 - x**2) + x*Derivative(y(x), (x, 2)) + sqrt(1 - x**2)) + y(x) - Derivative(y(x), (x, 2)))/x cannot be solved by the factorable group method