7.15.15 problem 15

Internal problem ID [471]
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 : 15
Date solved : Tuesday, March 04, 2025 at 11:24:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -2\right )^{2} y^{\prime \prime }-\left (x^{2}-4\right ) y^{\prime }+\left (x +2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

Order:=6; 
ode:=(x-2)^2*diff(diff(y(x),x),x)-(x^2-4)*diff(y(x),x)+(x+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{96} x^{4}+\frac {1}{192} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{96} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (\frac {x^5}{192}+\frac {x^4}{96}-\frac {x^3}{24}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {x^5}{96}+\frac {x^4}{48}-\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.951 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 2)**2*Derivative(y(x), (x, 2)) + (x + 2)*y(x) - (x**2 - 4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{96} - \frac {x^{3}}{24} - \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{3}}{48} - \frac {x^{2}}{12} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]