12.12.23 problem 25

Internal problem ID [1877]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 25
Date solved : Monday, January 27, 2025 at 05:37:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 4 \end{align*}

With initial conditions

\begin{align*} y \left (4\right )&=3\\ y^{\prime }\left (4\right )&=-4 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 20

Order:=6; 
dsolve([(x^2-8*x+14)*diff(y(x),x$2)-8*(x-4)*diff(y(x),x)+20*y(x)=0,y(4) = 3, D(y)(4) = -4],y(x),type='series',x=4);
 

\[ y = 3-4 \left (x -4\right )+15 \left (x -4\right )^{2}-4 \left (x -4\right )^{3}+\frac {15}{4} \left (x -4\right )^{4}-\frac {1}{5} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 44

AsymptoticDSolveValue[{(x^2-8*x+14)*D[y[x],{x,2}]+8*(x-4)*D[y[x],x]+20*y[x]==0,{y[4]==3,Derivative[1][y][4]==-4}},y[x],{x,4,"6"-1}]
 

\[ y(x)\to -\frac {35}{3} (x-4)^5+\frac {95}{4} (x-4)^4-\frac {28}{3} (x-4)^3+15 (x-4)^2-4 (x-4)+3 \]