6.15.38 problem 34

Internal problem ID [2036]
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 II. Exercises 7.6. Page 374
Problem number : 34
Date solved : Tuesday, September 30, 2025 at 05:22:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*(x^2+4)*diff(diff(y(x),x),x)+3*x*(3*x^2+8)*diff(y(x),x)+(-9*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (\ln \left (x \right ) c_2 +c_1 \right ) \left (1+\frac {5}{32} x^{2}-\frac {15}{2048} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {13}{64} x^{2}+\frac {13}{8192} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{{1}/{4}}} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 77

ode=4*x^2*(4+x^2)*D[y[x],{x,2}]+3*x*(8+3*x^2)*D[y[x],x]+(1-9*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {15 x^4}{2048}+\frac {5 x^2}{32}+1\right )}{\sqrt [4]{x}}+c_2 \left (\frac {\frac {13 x^4}{8192}-\frac {13 x^2}{64}}{\sqrt [4]{x}}+\frac {\left (-\frac {15 x^4}{2048}+\frac {5 x^2}{32}+1\right ) \log (x)}{\sqrt [4]{x}}\right ) \]

✓ Sympy. Time used: 0.500 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x**2 + 4)*Derivative(y(x), (x, 2)) + 3*x*(3*x**2 + 8)*Derivative(y(x), x) + (1 - 9*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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