12.13.13 problem 13

Internal problem ID [1904]
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 : 13
Date solved : Monday, January 27, 2025 at 05:38:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 18

Order:=6; 
dsolve([(1+x+2*x^2)*diff(y(x),x$2)+(1+7*x)*diff(y(x),x)+2*y(x)=0,y(1) = 1, D(y)(1) = 0],y(x),type='series',x=1);
 

\[ y = 1-\frac {1}{4} \left (x -1\right )^{2}+\frac {13}{48} \left (x -1\right )^{3}-\frac {77}{384} \left (x -1\right )^{4}+\frac {287}{2560} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 41

AsymptoticDSolveValue[{(1+x+2*x^2)*D[y[x],{x,2}]+(1+7*x)*D[y[x],x]+2*y[x]==0,{y[1]==1,Derivative[1][y][1]==0}},y[x],{x,1,"6"-1}]
 

\[ y(x)\to \frac {287 (x-1)^5}{2560}-\frac {77}{384} (x-1)^4+\frac {13}{48} (x-1)^3-\frac {1}{4} (x-1)^2+1 \]