6.15.4 problem Example 7.6.4 page 372

Internal problem ID [2002]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : Example 7.6.4 page 372
Date solved : Tuesday, September 30, 2025 at 05:22:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (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: 0.030 (sec). Leaf size: 40

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

\[ y = \left (\left (\left (-3\right ) x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}-\frac {1}{288} x^{4}+\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) x^{3} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 62

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

\[ y(x)\to c_1 (x+1) x^3+c_2 \left ((x+1) x^3 \log (x)+\left (\frac {x^5}{2400}-\frac {x^4}{288}+\frac {x^3}{36}-\frac {x^2}{4}-3 x\right ) x^3\right ) \]

✓ Sympy. Time used: 0.328 (sec). Leaf size: 14

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

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