6.15.46 problem 42

Internal problem ID [2044]
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 : 42
Date solved : Tuesday, September 30, 2025 at 05:22:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (-x^{2}+14\right ) y^{\prime }+2 \left (x^{2}+9\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*(x^2+2)*diff(diff(y(x),x),x)+x*(-x^2+14)*diff(y(x),x)+2*(x^2+9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {17}{8} x^{2}+\frac {85}{256} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {25}{8} x^{2}-\frac {471}{512} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{3}} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 71

ode=x^2*(2+x^2)*D[y[x],{x,2}]+x*(14-x^2)*D[y[x],x]+2*(9+x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (\frac {85 x^4}{256}-\frac {17 x^2}{8}+1\right )}{x^3}+c_2 \left (\frac {\frac {25 x^2}{8}-\frac {471 x^4}{512}}{x^3}+\frac {\left (\frac {85 x^4}{256}-\frac {17 x^2}{8}+1\right ) \log (x)}{x^3}\right ) \]

✓ Sympy. Time used: 0.458 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + x*(14 - x**2)*Derivative(y(x), x) + (2*x**2 + 18)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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