6.16.11 problem 7

Internal problem ID [2073]
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 III. Exercises 7.7. Page 389
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 05:23:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 60

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

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {1}{16} x +\frac {1}{640} x^{2}-\frac {1}{46080} x^{3}+\frac {1}{5160960} x^{4}-\frac {1}{825753600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {1}{64} x^{3}+\frac {1}{1024} x^{4}-\frac {1}{40960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {3}{16} x^{2}-\frac {5}{4096} x^{4}+\frac {39}{819200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 96

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

\[ y(x)\to c_2 \left (\frac {x^{11/2}}{5160960}-\frac {x^{9/2}}{46080}+\frac {x^{7/2}}{640}-\frac {x^{5/2}}{16}+x^{3/2}\right )+c_1 \left (\frac {(x-16) x^{3/2} \log (x)}{12288}-\frac {19 x^4-64 x^3-2304 x^2-18432 x-147456}{147456 x^{3/2}}\right ) \]

✓ Sympy. Time used: 0.272 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} x^{\frac {3}{2}} \left (- \frac {x^{3}}{46080} + \frac {x^{2}}{640} - \frac {x}{16} + 1\right ) + O\left (x^{6}\right ) \]