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 : Monday, January 27, 2025 at 04:56:21 AM
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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{(1-x)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==0,{y[0]==-3,Derivative[1][y][0] ==2}},y[x],{x,0,"6"-1}]
 

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