20.24.13 problem Problem 13

Internal problem ID [3998]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 13
Date solved : Sunday, March 30, 2025 at 02:14:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+4 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 54

Order:=6; 
ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+4*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}-\frac {2}{15} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {2}{3} x^{4}+\frac {7}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 61

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

\[ y(x)\to c_1 \left (-\frac {2 x^5}{15}+\frac {x^4}{3}-\frac {2 x^3}{3}+1\right )+c_2 \left (\frac {7 x^5}{15}-\frac {2 x^4}{3}+\frac {2 x^3}{3}-x^2+x\right ) \]

✓ Sympy. Time used: 0.812 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*y(x) + 2*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)
 

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