6.16.38 problem 34

Internal problem ID [2100]
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 : 34
Date solved : Tuesday, September 30, 2025 at 05:24:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 43

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(-3*x^2+1)*diff(y(x),x)-4*(-3*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (1944 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-648 x^{2}-810 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 50

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

\[ y(x)\to c_2 \left (x^2-\frac {x^4}{2}\right )+c_1 \left (\frac {18 x^4+9 x^2+2}{2 x^2}-\frac {27}{2} x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.353 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - 3*x**2)*Derivative(y(x), x) - (4 - 12*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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