12.16.35 problem 31

Internal problem ID [2097]
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 : 31
Date solved : Saturday, March 29, 2025 at 11:48:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+8*x*diff(y(x),x)-(-x^2+35)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{6} \left (1-\frac {1}{64} x^{2}+\frac {1}{10240} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-86400-2700 x^{2}-\frac {675}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{7}/{2}}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 58

ode=4*x^2*D[y[x],{x,2}]+8*x*D[y[x],x]-(35-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{32 x^{3/2}}+\frac {1}{x^{7/2}}+\frac {\sqrt {x}}{1024}\right )+c_2 \left (\frac {x^{13/2}}{10240}-\frac {x^{9/2}}{64}+x^{5/2}\right ) \]

✓ Sympy. Time used: 0.847 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 8*x*Derivative(y(x), x) - (35 - x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{\frac {5}{2}} \left (1 - \frac {x^{2}}{64}\right ) + O\left (x^{6}\right ) \]