44.20.1 problem 1

Internal problem ID [9415]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 06:18:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 56

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

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-4 x +4 x^{2}-\frac {16}{9} x^{3}+\frac {4}{9} x^{4}-\frac {16}{225} x^{5}+\frac {16}{2025} x^{6}-\frac {64}{99225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (8 x -12 x^{2}+\frac {176}{27} x^{3}-\frac {50}{27} x^{4}+\frac {1096}{3375} x^{5}-\frac {392}{10125} x^{6}+\frac {3872}{1157625} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) x^{2} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 158

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

\[ y(x)\to c_1 \left (-\frac {64 x^7}{99225}+\frac {16 x^6}{2025}-\frac {16 x^5}{225}+\frac {4 x^4}{9}-\frac {16 x^3}{9}+4 x^2-4 x+1\right ) x^2+c_2 \left (\left (\frac {3872 x^7}{1157625}-\frac {392 x^6}{10125}+\frac {1096 x^5}{3375}-\frac {50 x^4}{27}+\frac {176 x^3}{27}-12 x^2+8 x\right ) x^2+\left (-\frac {64 x^7}{99225}+\frac {16 x^6}{2025}-\frac {16 x^5}{225}+\frac {4 x^4}{9}-\frac {16 x^3}{9}+4 x^2-4 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.278 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {16 x^{5}}{225} + \frac {4 x^{4}}{9} - \frac {16 x^{3}}{9} + 4 x^{2} - 4 x + 1\right ) + O\left (x^{8}\right ) \]