67.26.5 problem 36.2 (e)

Internal problem ID [17032]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (e)
Date solved : Thursday, October 02, 2025 at 01:42:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (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*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{3} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+6 x +12 x^{2}+\frac {40}{3} x^{3}+10 x^{4}+\frac {28}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-10\right ) x -29 x^{2}-\frac {346}{9} x^{3}-\frac {193}{6} x^{4}-\frac {1459}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 112

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

\[ y(x)\to c_1 \left (\frac {28 x^5}{5}+10 x^4+\frac {40 x^3}{3}+12 x^2+6 x+1\right ) x^3+c_2 \left (\left (-\frac {1459 x^5}{75}-\frac {193 x^4}{6}-\frac {346 x^3}{9}-29 x^2-10 x\right ) x^3+\left (\frac {28 x^5}{5}+10 x^4+\frac {40 x^3}{3}+12 x^2+6 x+1\right ) x^3 \log (x)\right ) \]

✓ Sympy. Time used: 0.268 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} x^{3} \left (12 x^{2} + 6 x + 1\right ) + O\left (x^{6}\right ) \]