12.13.2 problem 2

Internal problem ID [1893]
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 : 2
Date solved : Monday, January 27, 2025 at 05:37:57 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+8 x \right ) y^{\prime }+4 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 )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 18

Order:=6; 
dsolve([(1+x+2*x^2)*diff(y(x),x$2)+(2+8*x)*diff(y(x),x)+4*y(x)=0,y(0) = -1, D(y)(0) = 2],y(x),type='series',x=0);
 

\[ y = -1+2 x -4 x^{3}+4 x^{4}+4 x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 23

AsymptoticDSolveValue[{(1+x+2*x^2)*D[y[x],{x,2}]+(2+8*x)*D[y[x],x]+4*y[x]==0,{y[0]==-1,Derivative[1][y][0] ==2}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to 4 x^5+4 x^4-4 x^3+2 x-1 \]