10.13.12 problem 15

Internal problem ID [1372]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number : 15
Date solved : Saturday, March 29, 2025 at 10:53:35 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

Order:=6; 
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 30

ode=D[y[x],{x,2}]-x*D[y[x],x]-y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {x^5}{15}+\frac {x^4}{4}+\frac {x^3}{3}+x^2+x+2 \]

✓ Sympy. Time used: 0.676 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) - y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]