6.13.40 problem 39 (b)

Internal problem ID [1931]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 39 (b)
Date solved : Tuesday, September 30, 2025 at 05:21:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

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

\[ y = x -x^{3}+\frac {1}{2} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 17

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

\[ y(x)\to \frac {x^5}{2}-x^3+x \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = - \frac {7 x^{5} r{\left (3 \right )}}{10} + C_{2} \left (\frac {x^{4}}{2} - x^{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{5}\right ) + O\left (x^{6}\right ) \]