12.13.17 problem 20

Internal problem ID [1908]
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 : 20
Date solved : Monday, January 27, 2025 at 05:38:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x +1\right ) y^{\prime \prime }-\left (1-2 x \right ) y^{\prime }-\left (3-2 x \right ) 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 )&=-2 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
dsolve([(1+2*x)*diff(y(x),x$2)-(1-2*x)*diff(y(x),x)-(3-2*x)*y(x)=0,y(1) = 1, D(y)(1) = -2],y(x),type='series',x=1);
 

\[ y = 1-2 \left (x -1\right )+\frac {1}{2} \left (x -1\right )^{2}-\frac {1}{6} \left (x -1\right )^{3}+\frac {5}{36} \left (x -1\right )^{4}-\frac {73}{1080} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{(1+2*x)*D[y[x],{x,2}]-(1-2*x)*D[y[x],x]-(3-2*x)*y[x]==0,{y[1]==1,Derivative[1][y][1]==-2}},y[x],{x,1,"6"-1}]
 

\[ y(x)\to -\frac {73 (x-1)^5}{1080}+\frac {5}{36} (x-1)^4-\frac {1}{6} (x-1)^3+\frac {1}{2} (x-1)^2-2 (x-1)+1 \]