12.14.34 problem 36

Internal problem ID [1975]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 36
Date solved : Tuesday, March 04, 2025 at 01:47:19 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 32

Order:=6; 
ode:=x*(x^2+3)*diff(diff(y(x),x),x)+(-x^2+2)*diff(y(x),x)-8*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1+\frac {11}{18} x^{2}+\frac {55}{648} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {4}{5} x^{2}+\frac {8}{55} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 47

ode=x*(3+x^2)*D[y[x],{x,2}]+(2-x^2)*D[y[x],x]-8*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {55 x^4}{648}+\frac {11 x^2}{18}+1\right )+c_2 \left (\frac {8 x^4}{55}+\frac {4 x^2}{5}+1\right ) \]

✓ Sympy. Time used: 1.232 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 3)*Derivative(y(x), (x, 2)) - 8*x*y(x) + (2 - x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {10368 x^{5}}{1925} + \frac {864 x^{4}}{55} + \frac {144 x^{3}}{5} + \frac {144 x^{2}}{5} + 12 x + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {1728 x^{4}}{455} + \frac {288 x^{3}}{35} + \frac {72 x^{2}}{7} + 6 x + 1\right ) + O\left (x^{6}\right ) \]