49.17.4 problem 1(d)

Internal problem ID [7711]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(d)
Date solved : Sunday, March 30, 2025 at 12:19:36 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 70

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-2 x +\frac {4}{3} x^{2}-\frac {4}{9} x^{3}+\frac {4}{45} x^{4}-\frac {8}{675} x^{5}+\frac {16}{14175} x^{6}-\frac {8}{99225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\left (-4\right ) x +8 x^{2}-\frac {16}{3} x^{3}+\frac {16}{9} x^{4}-\frac {16}{45} x^{5}+\frac {32}{675} x^{6}-\frac {64}{14175} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-12 x^{2}+\frac {112}{9} x^{3}-\frac {140}{27} x^{4}+\frac {808}{675} x^{5}-\frac {1792}{10125} x^{6}+\frac {9056}{496125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 119

ode=x*D[y[x],{x,2}]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {4}{675} x \left (8 x^5-60 x^4+300 x^3-900 x^2+1350 x-675\right ) \log (x)+\frac {-2272 x^6+15720 x^5-70500 x^4+180000 x^3-202500 x^2+40500 x+10125}{10125}\right )+c_2 \left (\frac {16 x^7}{14175}-\frac {8 x^6}{675}+\frac {4 x^5}{45}-\frac {4 x^4}{9}+\frac {4 x^3}{3}-2 x^2+x\right ) \]

✓ Sympy. Time used: 0.787 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {16 x^{6}}{14175} - \frac {8 x^{5}}{675} + \frac {4 x^{4}}{45} - \frac {4 x^{3}}{9} + \frac {4 x^{2}}{3} - 2 x + 1\right ) + O\left (x^{8}\right ) \]