20.24.14 problem Problem 14

Internal problem ID [3999]
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 14
Date solved : Tuesday, March 04, 2025 at 05:22:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 54

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

\[ y \left (x \right ) = \left (1-x^{2}-\frac {1}{6} x^{3}+\frac {1}{3} x^{4}+\frac {11}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{12} x^{4}+\frac {1}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

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

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

\[ y(x)\to c_2 \left (\frac {x^5}{8}-\frac {x^4}{12}-\frac {x^3}{2}+x\right )+c_1 \left (\frac {11 x^5}{120}+\frac {x^4}{3}-\frac {x^3}{6}-x^2+1\right ) \]

✓ Sympy. Time used: 0.828 (sec). Leaf size: 37

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

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