12.13.39 problem 39 (a)

Internal problem ID [1930]
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 (a)
Date solved : Monday, January 27, 2025 at 05:38:35 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 )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{D[y[x],{x,2}]+4*x*D[y[x],x]+(2+4*x^2)*y[x]==0,{y[0]==1,Derivative[1][y][0] ==0}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {x^4}{2}-x^2+1 \]