6.16.37 problem 33

Internal problem ID [2099]
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 : 33
Date solved : Tuesday, September 30, 2025 at 05:23:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (-x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 41

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(-2*x^2+1)*diff(y(x),x)-4*(-x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (288 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-288 x^{2}-216 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 37

ode=x^2*D[y[x],{x,2}]+x*(1-2*x^2)*D[y[x],x]-4*(1-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.338 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - 2*x**2)*Derivative(y(x), x) - (4 - 4*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{2} + O\left (x^{6}\right ) \]