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 : Monday, January 27, 2025 at 02:31:12 PM
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*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

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

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

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