67.25.23 problem 35.4 (i)

Internal problem ID [17018]
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 (i)
Date solved : Thursday, October 02, 2025 at 01:41:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+\left (9+4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-(2*x^2+5*x)*diff(y(x),x)+(9+4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {22}{9} x^{3}-\frac {25}{18} x^{4}-\frac {137}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{3} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 116

ode=x^2*D[y[x],{x,2}]-(5*x+2*x^2)*D[y[x],x]+(9+4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+2 x+1\right ) x^3+c_2 \left (\left (-\frac {137 x^5}{225}-\frac {25 x^4}{18}-\frac {22 x^3}{9}-3 x^2-2 x\right ) x^3+\left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+2 x+1\right ) x^3 \log (x)\right ) \]

✓ Sympy. Time used: 0.337 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (4*x + 9)*y(x) - (2*x**2 + 5*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{3} \left (2 x^{2} + 2 x + 1\right ) + O\left (x^{6}\right ) \]