9.8.13 problem problem 13

Internal problem ID [1078]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 13
Date solved : Saturday, March 29, 2025 at 10:38:22 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 29

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

\[ y = \left (1-\frac {x^{3}}{3}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

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

\[ y(x)\to c_2 \left (x-\frac {x^4}{4}\right )+c_1 \left (1-\frac {x^3}{3}\right ) \]

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

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + 2*x*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)
 

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