6.16.12 problem 8

Internal problem ID [2074]
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 : 8
Date solved : Tuesday, September 30, 2025 at 05:23:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+10 x y^{\prime }+\left (14+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 56

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+10*x*diff(y(x),x)+(14+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{6} x +\frac {1}{84} x^{2}-\frac {1}{2016} x^{3}+\frac {1}{72576} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{5}+c_2 \left (\ln \left (x \right ) \left (-x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2880+720 x +120 x^{2}+20 x^{3}+5 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{7}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 68

ode=x^2*D[y[x],{x,2}]+10*x*D[y[x],x]+(14+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^2}{72576}+\frac {1}{x^2}-\frac {x}{2016}-\frac {1}{6 x}+\frac {1}{84}\right )+c_1 \left (\frac {1}{x^7}+\frac {1}{4 x^6}+\frac {1}{24 x^5}+\frac {1}{144 x^4}+\frac {1}{576 x^3}\right ) \]

✓ Sympy. Time used: 0.268 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 10*x*Derivative(y(x), x) + (x + 14)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (- \frac {x^{7}}{20118067200} + \frac {x^{6}}{239500800} - \frac {x^{5}}{3628800} + \frac {x^{4}}{72576} - \frac {x^{3}}{2016} + \frac {x^{2}}{84} - \frac {x}{6} + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]