49.15.4 problem 1(d)

Internal problem ID [7690]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 1(d)
Date solved : Wednesday, March 05, 2025 at 04:50:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{3} y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

Order:=6; 
ode:=diff(diff(y(x),x),x)+x^3*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 28

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

\[ y(x)\to c_2 \left (x-\frac {x^5}{10}\right )+c_1 \left (1-\frac {x^4}{12}\right ) \]

✓ Sympy. Time used: 0.760 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{4}}{12}\right ) + C_{1} x \left (1 - \frac {x^{4}}{10}\right ) + O\left (x^{6}\right ) \]