54.6.7 problem 7

Internal problem ID [8639]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 7
Date solved : Sunday, March 30, 2025 at 01:22:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 53

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {1}{2} x +\frac {3}{20} x^{2}-\frac {1}{30} x^{3}+\frac {1}{168} x^{4}-\frac {1}{1120} x^{5}+\frac {1}{8640} x^{6}-\frac {1}{75600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (12-6 x +x^{3}-\frac {1}{2} x^{4}+\frac {3}{20} x^{5}-\frac {1}{30} x^{6}+\frac {1}{168} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 91

ode=x^2*D[y[x],{x,2}]+x^2*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{360}+\frac {x^4}{80}-\frac {x^3}{24}+\frac {x^2}{12}+\frac {1}{x}-\frac {1}{2}\right )+c_2 \left (\frac {x^8}{8640}-\frac {x^7}{1120}+\frac {x^6}{168}-\frac {x^5}{30}+\frac {3 x^4}{20}-\frac {x^3}{2}+x^2\right ) \]

✓ Sympy. Time used: 0.822 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {x^{5}}{1120} + \frac {x^{4}}{168} - \frac {x^{3}}{30} + \frac {3 x^{2}}{20} - \frac {x}{2} + 1\right ) + \frac {C_{1} \left (\frac {x}{2} - 1\right )}{x} + O\left (x^{8}\right ) \]