6.16.32 problem 28

Internal problem ID [2094]
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 : 28
Date solved : Tuesday, September 30, 2025 at 05:23:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+8\right ) y^{\prime }+\left (3 x^{2}+5\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 48

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+2*x*(x^2+8)*diff(y(x),x)+(3*x^2+5)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{16} x^{2}+\frac {1}{256} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_2 \left (\ln \left (x \right ) \left (-\frac {1}{2} x^{2}+\frac {1}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {1}{2} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 72

ode=4*x^2*D[y[x],{x,2}]+2*x*(8+x^2)*D[y[x],x]+(5+3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^{7/2}}{256}-\frac {x^{3/2}}{16}+\frac {1}{\sqrt {x}}\right )+c_1 \left (\frac {5 x^4-96 x^2+256}{256 x^{5/2}}-\frac {\left (x^2-16\right ) \log (x)}{64 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.324 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 2*x*(x**2 + 8)*Derivative(y(x), x) + (3*x**2 + 5)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {x^{4}}{256} - \frac {x^{2}}{16} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]