6.15.56 problem 52

Internal problem ID [2054]
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 : 52
Date solved : Tuesday, September 30, 2025 at 05:23:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 36

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+2*x*(-x^2+4)*diff(y(x),x)+(7*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{2} x^{2}+\frac {1}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {5}{8} x^{2}-\frac {9}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 77

ode=4*x^2*D[y[x],{x,2}]+2*x*(4-x^2)*D[y[x],x]+(1+7*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (\frac {x^4}{32}-\frac {x^2}{2}+1\right )}{\sqrt {x}}+c_2 \left (\frac {\frac {5 x^2}{8}-\frac {9 x^4}{128}}{\sqrt {x}}+\frac {\left (\frac {x^4}{32}-\frac {x^2}{2}+1\right ) \log (x)}{\sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.363 (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*(4 - x**2)*Derivative(y(x), x) + (7*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {x^{4}}{32} - \frac {x^{2}}{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]