18.3.2 problem Problem 16.2

Internal problem ID [3502]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 16, Series solutions of ODEs. Section 16.6 Exercises, page 550
Problem number : Problem 16.2
Date solved : Monday, January 27, 2025 at 07:39:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 85

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

\[ y(z)\to c_1 \sqrt {z} \left (\frac {z^5}{10395}+\frac {z^4}{945}+\frac {z^3}{105}+\frac {z^2}{15}+\frac {z}{3}+1\right )+c_2 \left (\frac {z^5}{3840}+\frac {z^4}{384}+\frac {z^3}{48}+\frac {z^2}{8}+\frac {z}{2}+1\right ) \]