46.5.9 problem 19

Internal problem ID [7345]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. REVIEW QUESTIONS. page 201
Problem number : 19
Date solved : Wednesday, March 05, 2025 at 04:23:46 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 58

Order:=6; 
ode:=diff(diff(y(x),x),x)+1/4*y(x)/x = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x \left (1-\frac {1}{8} x +\frac {1}{192} x^{2}-\frac {1}{9216} x^{3}+\frac {1}{737280} x^{4}-\frac {1}{88473600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{4} x +\frac {1}{32} x^{2}-\frac {1}{768} x^{3}+\frac {1}{36864} x^{4}-\frac {1}{2949120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{64} x^{2}+\frac {7}{2304} x^{3}-\frac {35}{442368} x^{4}+\frac {101}{88473600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 85

ode=D[y[x],{x,2}]+1/(4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x \left (x^3-48 x^2+1152 x-9216\right ) \log (x)}{36864}+\frac {-47 x^4+1920 x^3-34560 x^2+110592 x+442368}{442368}\right )+c_2 \left (\frac {x^5}{737280}-\frac {x^4}{9216}+\frac {x^3}{192}-\frac {x^2}{8}+x\right ) \]

✓ Sympy. Time used: 0.867 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + y(x)/(4*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{737280} - \frac {x^{3}}{9216} + \frac {x^{2}}{192} - \frac {x}{8} + 1\right ) + O\left (x^{6}\right ) \]