10.13.15 problem 18

Internal problem ID [1375]
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 : 18
Date solved : Saturday, March 29, 2025 at 10:53:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-x \right ) 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 )&=-3\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

Order:=6; 
ode:=(1-x)*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = 0; 
ic:=y(0) = -3, D(y)(0) = 2; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = -3+2 x -\frac {3}{2} x^{2}-\frac {1}{2} x^{3}-\frac {1}{8} x^{4}-\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

ode=(1-x)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==0; 
ic={y[0]==-3,Derivative[1][y][0] ==2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {x^5}{40}-\frac {x^4}{8}-\frac {x^3}{2}-\frac {3 x^2}{2}+2 x-3 \]

✓ Sympy. Time used: 0.783 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + (1 - x)*Derivative(y(x), (x, 2)) - y(x),0) 
ics = {y(0): -3, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + 1\right ) + C_{1} x + O\left (x^{6}\right ) \]