23.5.8 problem 3(g)

Internal problem ID [4185]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(g)
Date solved : Monday, January 27, 2025 at 08:41:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 44

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

\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 86

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

\[ \{y(x)\}\to c_2 \left (\frac {627 x^5}{4259840}+\frac {33 x^4}{16384}+\frac {11 x^3}{512}+\frac {21 x^2}{128}+\frac {3 x}{4}+1\right )+c_1 x^{2/3} \left (\frac {819221 x^5}{36787322880}+\frac {1071289 x^4}{3065610240}+\frac {20213 x^3}{4561920}+\frac {493 x^2}{11520}+\frac {17 x}{60}+1\right ) \]