12.16.44 problem 40

Internal problem ID [2106]
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 : 40
Date solved : Tuesday, March 04, 2025 at 01:50:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*(x^2+1)*diff(diff(y(x),x),x)+4*x*(x^2+2)*diff(y(x),x)-(x^2+15)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {1}{6} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-144-216 x^{2}-54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 58

ode=4*x^2*(1+x^2)*D[y[x],{x,2}]+4*x*(2+x^2)*D[y[x],x]-(15+x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {3 x^{3/2}}{8}+\frac {1}{x^{5/2}}+\frac {3}{2 \sqrt {x}}\right )+c_2 \left (\frac {x^{11/2}}{16}-\frac {x^{7/2}}{6}+x^{3/2}\right ) \]

✓ Sympy. Time used: 1.114 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 4*x*(x**2 + 2)*Derivative(y(x), x) - (x**2 + 15)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} + \frac {C_{1}}{x^{\frac {5}{2}}} + O\left (x^{6}\right ) \]