73.25.26 problem 35.4 (L)

Internal problem ID [15663]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (L)
Date solved : Monday, March 31, 2025 at 01:44:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+8 x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 46

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+8*x^2*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {5}{12} x^{3}+\frac {35}{192} x^{4}-\frac {21}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {19}{128} x^{4}+\frac {25}{384} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 125

ode=4*x^2*D[y[x],{x,2}]+8*x^2*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right )+c_2 \left (\sqrt {x} \left (\frac {25 x^5}{384}-\frac {19 x^4}{128}+\frac {x^3}{4}-\frac {x^2}{4}\right )+\sqrt {x} \left (-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.792 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {35 x^{4}}{192} - \frac {5 x^{3}}{12} + \frac {3 x^{2}}{4} - x + 1\right ) + O\left (x^{6}\right ) \]