45.2.33 problem 33

Internal problem ID [7256]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 33
Date solved : Sunday, March 30, 2025 at 11:52:37 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 54

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

\[ y(x)\to c_1 x e^{-1+\frac {i \sqrt {\lambda }}{x}}-\frac {i c_2 x e^{1-\frac {i \sqrt {\lambda }}{x}}}{2 \sqrt {\lambda }} \]

✗ Sympy

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

ValueError : ODE lambda_*y(x) + x**4*Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular