46.5.8 problem 18

Internal problem ID [7344]
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 : 18
Date solved : Wednesday, March 05, 2025 at 04:23:45 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 28

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+3*diff(y(x),x)+4*x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 30

ode=x*D[y[x],{x,2}]+3*D[y[x],x]+4*x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.912 (sec). Leaf size: 26

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

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