78.3.9 problem 2.d

Internal problem ID [21005]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 2.d
Date solved : Thursday, October 02, 2025 at 07:01:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

Order:=5; 
ode:=diff(diff(y(x),x),x)+2*x^2*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{5}\right ) \]

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

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

\[ y(x)\to c_2 \left (x-\frac {x^4}{4}\right )+c_1 \left (1-\frac {x^3}{6}\right ) \]

✓ Sympy. Time used: 0.219 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{20} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {x^{4}}{10} - \frac {x^{3}}{12} - \frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]