6.12.22 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 : Tuesday, September 30, 2025 at 05:20:48 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*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

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

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

ode=(4*x^2-24*x+37)*D[y[x],{x,2}]+y[x]==0; 
ic={y[3]==4,Derivative[1][y][3 ]==-6}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
 

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

✓ Sympy. Time used: 0.246 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*x**2 - 24*x + 37)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {y(3): 4, Subs(Derivative(y(x), x), x, 3): -6} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=3,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {3 \left (x - 3\right )^{4}}{8} - \frac {\left (x - 3\right )^{2}}{2} + 1\right ) + C_{1} \left (x - \frac {\left (x - 3\right )^{3}}{6} - 3\right ) + O\left (x^{6}\right ) \]