10.14.9 problem 6. case \(x_0=-4\)

Internal problem ID [1391]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 6. case \(x_0=-4\)
Date solved : Monday, January 27, 2025 at 04:56:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-2 x -3\right ) y^{\prime \prime }+x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -4 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 76

Order:=6; 
dsolve((x^2-2*x-3)*diff(y(x),x$2)+x*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=-4);
 

\[ y = \left (1-\frac {2 \left (x +4\right )^{2}}{21}-\frac {4 \left (x +4\right )^{3}}{189}-\frac {4 \left (x +4\right )^{4}}{1323}-\frac {\left (x +4\right )^{5}}{3087}\right ) y \left (-4\right )+\left (x +4+\frac {2 \left (x +4\right )^{2}}{21}-\frac {\left (x +4\right )^{3}}{54}-\frac {11 \left (x +4\right )^{4}}{1323}-\frac {157 \left (x +4\right )^{5}}{74088}\right ) y^{\prime }\left (-4\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 87

AsymptoticDSolveValue[(x^2-2*x-3)*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==0,y[x],{x,-4,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {(x+4)^5}{3087}-\frac {4 (x+4)^4}{1323}-\frac {4}{189} (x+4)^3-\frac {2}{21} (x+4)^2+1\right )+c_2 \left (-\frac {157 (x+4)^5}{74088}-\frac {11 (x+4)^4}{1323}-\frac {1}{54} (x+4)^3+\frac {2}{21} (x+4)^2+x+4\right ) \]