6.15.16 problem 12

Internal problem ID [2014]
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 : 12
Date solved : Tuesday, September 30, 2025 at 05:22:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 48

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

\[ y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 128

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right )+c_2 \left (\sqrt {x} \left (\frac {137 x^5}{432000}-\frac {25 x^4}{3456}+\frac {11 x^3}{108}-\frac {3 x^2}{4}+2 x\right )+\sqrt {x} \left (-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.245 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {x^{4}}{576} - \frac {x^{3}}{36} + \frac {x^{2}}{4} - x + 1\right ) + O\left (x^{6}\right ) \]