12.13.50 problem 49

Internal problem ID [1941]
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 : 49
Date solved : Monday, January 27, 2025 at 05:38:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}-11 x +16\right ) y^{\prime \prime }+\left (x^{2}-6 x +10\right ) y^{\prime }-\left (2-x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

With initial conditions

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

✓ Solution by Maple

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

Order:=6; 
dsolve([(16-11*x+2*x^2)*diff(y(x),x$2)+(10-6*x+x^2)*diff(y(x),x)-(2-x)*y(x)=0,y(3) = 1, D(y)(3) = -2],y(x),type='series',x=3);
 

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 46

AsymptoticDSolveValue[{(16-11*x+2*x^2)*D[y[x],{x,2}]+(10-6*x+x^2)*D[y[x],x]-(2-x)*y[x]==0,{y[3]==1,Derivative[1][y][3 ]==-2}},y[x],{x,3,"6"-1}]
 

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