42.1.13 problem 3.24 (h)

Internal problem ID [6835]
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 : 3.24 (h)
Date solved : Monday, January 27, 2025 at 02:31:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x \left (x +2\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 38

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

\[ y = c_1 \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {7}{32} x^{2}-\frac {3}{128} x^{3}+\frac {11}{2048} x^{4}-\frac {13}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+4 x +2 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 60

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

\[ y(x)\to c_2 \left (2 x^2+4 x+1\right )+c_1 \sqrt {x} \left (-\frac {13 x^5}{8192}+\frac {11 x^4}{2048}-\frac {3 x^3}{128}+\frac {7 x^2}{32}+\frac {5 x}{4}+1\right ) \]