Internal problem ID [2022]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(z = 0\).
✓ Solution by Maple
Time used: 0.031 (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);
\[ y \relax (z ) = c_{1} \sqrt {z}\, \left (1+\frac {1}{3} z +\frac {1}{15} z^{2}+\frac {1}{105} z^{3}+\frac {1}{945} z^{4}+\frac {1}{10395} z^{5}+\mathrm {O}\left (z^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} z +\frac {1}{8} z^{2}+\frac {1}{48} z^{3}+\frac {1}{384} z^{4}+\frac {1}{3840} z^{5}+\mathrm {O}\left (z^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 85
AsymptoticDSolveValue[4*z*y''[z]+2*(1-z)*y'[z]-y[z]==0,y[z],{z,0,5}]
\[ y(z)\to c_1 \sqrt {z} \left (\frac {z^5}{10395}+\frac {z^4}{945}+\frac {z^3}{105}+\frac {z^2}{15}+\frac {z}{3}+1\right )+c_2 \left (\frac {z^5}{3840}+\frac {z^4}{384}+\frac {z^3}{48}+\frac {z^2}{8}+\frac {z}{2}+1\right ) \]