12.16.31 problem 27

Internal problem ID [2093]
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 : 27
Date solved : Saturday, March 29, 2025 at 11:48:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 44

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

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 52

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

\[ y(x)\to c_1 \left (\frac {\left (x^2+8\right )^2}{64 x}-\frac {1}{16} x^3 \log (x)\right )+c_2 \left (\frac {x^7}{384}-\frac {x^5}{12}+x^3\right ) \]

✓ Sympy. Time used: 0.861 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} x^{3} \left (1 - \frac {x^{2}}{12}\right ) + O\left (x^{6}\right ) \]