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 : Monday, January 27, 2025 at 04:56:18 AM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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