12.13.10 problem 10

Internal problem ID [1901]
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 : 10
Date solved : Monday, January 27, 2025 at 05:38:05 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

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

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

\[ y = 2-\left (x -1\right )-\frac {1}{2} \left (x -1\right )^{2}+\frac {5}{3} \left (x -1\right )^{3}-\frac {19}{12} \left (x -1\right )^{4}+\frac {7}{30} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 44

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

\[ y(x)\to \frac {7}{30} (x-1)^5-\frac {19}{12} (x-1)^4+\frac {5}{3} (x-1)^3-\frac {1}{2} (x-1)^2-x+3 \]