40.2.15 problem 17

Internal problem ID [8604]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:39:40 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x y^{\prime \prime }+y^{\prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 44

Order:=6; 
ode:=4*x*diff(diff(y(x),x),x)+diff(y(x),x)+8*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{3}/{4}} \left (1-\frac {8}{7} x +\frac {32}{77} x^{2}-\frac {256}{3465} x^{3}+\frac {512}{65835} x^{4}-\frac {4096}{7571025} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-8 x +\frac {32}{5} x^{2}-\frac {256}{135} x^{3}+\frac {512}{1755} x^{4}-\frac {4096}{149175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 83

ode=4*x*D[y[x],{x,2}]+D[y[x],x]+8*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {4096 x^5}{149175}+\frac {512 x^4}{1755}-\frac {256 x^3}{135}+\frac {32 x^2}{5}-8 x+1\right )+c_1 x^{3/4} \left (-\frac {4096 x^5}{7571025}+\frac {512 x^4}{65835}-\frac {256 x^3}{3465}+\frac {32 x^2}{77}-\frac {8 x}{7}+1\right ) \]

✓ Sympy. Time used: 0.266 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) + 8*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {4096 x^{5}}{149175} + \frac {512 x^{4}}{1755} - \frac {256 x^{3}}{135} + \frac {32 x^{2}}{5} - 8 x + 1\right ) + C_{1} x^{\frac {3}{4}} \left (\frac {512 x^{4}}{65835} - \frac {256 x^{3}}{3465} + \frac {32 x^{2}}{77} - \frac {8 x}{7} + 1\right ) + O\left (x^{6}\right ) \]