42.1.1 problem 3.5

Internal problem ID [6823]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.5
Date solved : Sunday, March 30, 2025 at 11:23:45 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x -1\right ) \left (x -2\right ) y^{\prime \prime }+\left (4 x -6\right ) y^{\prime }+2 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: 0.005 (sec). Leaf size: 20

Order:=6; 
ode:=(x-1)*(x-2)*diff(diff(y(x),x),x)+(4*x-6)*diff(y(x),x)+2*y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 2+x +\frac {1}{2} x^{2}+\frac {1}{4} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

ode=(x-1)*(x-2)*D[y[x],{x,2}]+(4*x-6)*D[y[x],x]+2*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {x^5}{16}+\frac {x^4}{8}+\frac {x^3}{4}+\frac {x^2}{2}+x+2 \]

✓ Sympy. Time used: 1.139 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 2)*(x - 1)*Derivative(y(x), (x, 2)) + (4*x - 6)*Derivative(y(x), x) + 2*y(x),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)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {7 x^{4}}{8} - \frac {3 x^{3}}{4} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {15 x^{3}}{8} + \frac {7 x^{2}}{4} + \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]