12.12.22 problem 24

Internal problem ID [1876]
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 : 24
Date solved : Monday, January 27, 2025 at 05:37:41 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (4 x^{2}-24 x +37\right ) y^{\prime \prime }+y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (3\right )&=4\\ y^{\prime }\left (3\right )&=-6 \end{align*}

✓ Solution by Maple

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

Order:=6; 
dsolve([(4*x^2-24*x+37)*diff(y(x),x$2)+y(x)=0,y(3) = 4, D(y)(3) = -6],y(x),type='series',x=3);
 

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

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

AsymptoticDSolveValue[{(4*x^2-24*x+37)*D[y[x],{x,2}]+y[x]==0,{y[3]==4,Derivative[1][y][3 ]==-6}},y[x],{x,3,"6"-1}]
 

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