10.14.7 problem 6. case \(x_0=0\)

Internal problem ID [1389]
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 \(x_0=0\)
Date solved : Monday, January 27, 2025 at 04:56:34 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*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

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=0);
 

\[ y = \left (1+\frac {2}{3} x^{2}-\frac {4}{27} x^{3}+\frac {16}{81} x^{4}-\frac {1}{9} x^{5}\right ) y \left (0\right )+\left (x +\frac {5}{18} x^{3}-\frac {5}{54} x^{4}+\frac {7}{72} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (\frac {7 x^5}{72}-\frac {5 x^4}{54}+\frac {5 x^3}{18}+x\right )+c_1 \left (-\frac {x^5}{9}+\frac {16 x^4}{81}-\frac {4 x^3}{27}+\frac {2 x^2}{3}+1\right ) \]