54.5.3 problem 3

Internal problem ID [8618]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 3
Date solved : Sunday, March 30, 2025 at 01:21:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 56

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+x*(x-3)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {3}{2} x^{2}-\frac {2}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{20} x^{5}+\frac {7}{720} x^{6}-\frac {1}{630} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {13}{4} x^{2}+\frac {31}{18} x^{3}-\frac {173}{288} x^{4}+\frac {187}{1200} x^{5}-\frac {463}{14400} x^{6}+\frac {971}{176400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) x^{2} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 164

ode=x^2*D[y[x],{x,2}]+x*(x-3)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^7}{630}+\frac {7 x^6}{720}-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right ) x^2+c_2 \left (\left (\frac {971 x^7}{176400}-\frac {463 x^6}{14400}+\frac {187 x^5}{1200}-\frac {173 x^4}{288}+\frac {31 x^3}{18}-\frac {13 x^2}{4}+3 x\right ) x^2+\left (-\frac {x^7}{630}+\frac {7 x^6}{720}-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.884 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(x - 3)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {x^{5}}{20} + \frac {5 x^{4}}{24} - \frac {2 x^{3}}{3} + \frac {3 x^{2}}{2} - 2 x + 1\right ) + O\left (x^{8}\right ) \]