6.15.12 problem 8

Internal problem ID [2010]
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 : 8
Date solved : Tuesday, September 30, 2025 at 05:22:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 54

Order:=8; 
ode:=x^2*(2*x+1)*diff(diff(y(x),x),x)+x*(3*x^2+14*x+5)*diff(y(x),x)+(12*x^2+18*x+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {5}{2} x^{2}-3 x^{3}+\frac {33}{8} x^{4}-\frac {129}{20} x^{5}+\frac {867}{80} x^{6}-\frac {1059}{56} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{4} x^{2}-\frac {13}{6} x^{3}+\frac {407}{96} x^{4}-\frac {9047}{1200} x^{5}+\frac {63851}{4800} x^{6}-\frac {559033}{23520} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x^{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 157

ode=x^2*(1+2*x)*D[y[x],{x,2}]+x*(5+14*x+3*x^2)*D[y[x],x]+(4+18*x+12*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {1059 x^7}{56}+\frac {867 x^6}{80}-\frac {129 x^5}{20}+\frac {33 x^4}{8}-3 x^3+\frac {5 x^2}{2}-2 x+1\right )}{x^2}+c_2 \left (\frac {-\frac {559033 x^7}{23520}+\frac {63851 x^6}{4800}-\frac {9047 x^5}{1200}+\frac {407 x^4}{96}-\frac {13 x^3}{6}+\frac {3 x^2}{4}}{x^2}+\frac {\left (-\frac {1059 x^7}{56}+\frac {867 x^6}{80}-\frac {129 x^5}{20}+\frac {33 x^4}{8}-3 x^3+\frac {5 x^2}{2}-2 x+1\right ) \log (x)}{x^2}\right ) \]

✓ Sympy. Time used: 0.514 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) + x*(3*x**2 + 14*x + 5)*Derivative(y(x), x) + (12*x**2 + 18*x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x^{2}} + O\left (x^{8}\right ) \]