12.15.40 problem 36

Internal problem ID [2038]
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 : 36
Date solved : Saturday, March 29, 2025 at 11:46:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (4 x^{2}+1\right ) y^{\prime \prime }+32 x^{3} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*(4*x^2+1)*diff(diff(y(x),x),x)+32*x^3*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {3}{4} x^{2}+\frac {105}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {5}{4} x^{2}+\frac {389}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 77

ode=4*x^2*(1+4*x^2)*D[y[x],{x,2}]+32*x^3*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {105 x^4}{64}-\frac {3 x^2}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {389 x^4}{128}-\frac {5 x^2}{4}\right )+\sqrt {x} \left (\frac {105 x^4}{64}-\frac {3 x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 1.005 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(32*x**3*Derivative(y(x), x) + 4*x**2*(4*x**2 + 1)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {140 x^{4}}{3} - \frac {80 x^{3}}{3} + 12 x^{2} - 4 x + 1\right ) + O\left (x^{6}\right ) \]