42.1.14 problem 3.24 (i)

Internal problem ID [6836]
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 (i)
Date solved : Monday, January 27, 2025 at 02:31:26 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x y^{\prime \prime }+\left (\frac {1}{2}-x \right ) y^{\prime }-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: 44

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

\[ y = c_1 \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+2 x +\frac {4}{3} x^{2}+\frac {8}{15} x^{3}+\frac {16}{105} x^{4}+\frac {32}{945} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 79

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (\frac {32 x^5}{945}+\frac {16 x^4}{105}+\frac {8 x^3}{15}+\frac {4 x^2}{3}+2 x+1\right ) \]