7.15.40 problem 41

Internal problem ID [496]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 41
Date solved : Tuesday, March 04, 2025 at 11:25:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x -1\right ) \left (x +1\right )^{2} y^{\prime \prime }+2 x \left (x -3\right ) \left (x +1\right ) y^{\prime }-2 \left (x -1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 60

Order:=6; 
ode:=x*(x-1)*(1+x)^2*diff(diff(y(x),x),x)+2*x*(x-3)*(1+x)*diff(y(x),x)-2*(x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-2 x +3 x^{2}-4 x^{3}+5 x^{4}-6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2 x -4 x^{2}+6 x^{3}-8 x^{4}+10 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1-6 x +10 x^{2}-14 x^{3}+18 x^{4}-22 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 58

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^{7/2}}{24}-\frac {x^{3/2}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{9/2}}{120}-\frac {x^{5/2}}{6}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 1.165 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(x - 3)*(x + 1)*Derivative(y(x), x) + x*(x - 1)*(x + 1)**2*Derivative(y(x), (x, 2)) - (2*x - 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{180} + \frac {x^{3}}{18} + \frac {x^{2}}{3} + x + 1\right ) + O\left (x^{6}\right ) \]