74.16.1 problem 1

Internal problem ID [16436]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 1
Date solved : Monday, March 31, 2025 at 02:53:20 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+7 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 60

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+7*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1-\frac {7 \left (x -1\right )^{2}}{2}+\frac {35 \left (x -1\right )^{4}}{24}-\frac {7 \left (x -1\right )^{5}}{6}\right ) y \left (1\right )+\left (x -1+\left (x -1\right )^{2}-\frac {5 \left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{5}}{24}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 65

ode=x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+7*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (-\frac {7}{6} (x-1)^5+\frac {35}{24} (x-1)^4-\frac {7}{2} (x-1)^2+1\right )+c_2 \left (\frac {7}{24} (x-1)^5-\frac {5}{6} (x-1)^3+(x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.711 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 2*x*Derivative(y(x), x) + 7*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {5 \left (x - 1\right )^{3}}{6} + \left (x - 1\right )^{2} - 1\right ) + C_{1} \left (\frac {35 \left (x - 1\right )^{4}}{24} - \frac {7 \left (x - 1\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]