12.13.23 problem 26

Internal problem ID [1914]
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 : 26
Date solved : Monday, January 27, 2025 at 05:38:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (10-2 x \right ) y^{\prime \prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

Order:=6; 
dsolve([(10-2*x)*diff(y(x),x$2)+(1+x)*y(x)=0,y(2) = 2, D(y)(2) = -4],y(x),type='series',x=2);
 

\[ y = 2-4 \left (x -2\right )-\frac {1}{2} \left (x -2\right )^{2}+\frac {2}{9} \left (x -2\right )^{3}+\frac {49}{432} \left (x -2\right )^{4}+\frac {23}{1080} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 46

AsymptoticDSolveValue[{(10-2*x)*D[y[x],{x,2}]+(1+x)*y[x]==0,{y[2]==2,Derivative[1][y][2]==-4}},y[x],{x,2,"6"-1}]
 

\[ y(x)\to \frac {23 (x-2)^5}{1080}+\frac {49}{432} (x-2)^4+\frac {2}{9} (x-2)^3-\frac {1}{2} (x-2)^2-4 (x-2)+2 \]