Internal problem ID [2025]
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 16.6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (z +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(z = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 47
Order:=6; dsolve(z^2*diff(y(z),z$2)-3/2*z*diff(y(z),z)+(1+z)*y(z)=0,y(z),type='series',z=0);
\[ y \relax (z ) = c_{1} \sqrt {z}\, \left (1+2 z -2 z^{2}+\frac {4}{9} z^{3}-\frac {2}{45} z^{4}+\frac {4}{1575} z^{5}+\mathrm {O}\left (z^{6}\right )\right )+c_{2} z^{2} \left (1-\frac {2}{5} z +\frac {2}{35} z^{2}-\frac {4}{945} z^{3}+\frac {2}{10395} z^{4}-\frac {4}{675675} z^{5}+\mathrm {O}\left (z^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 84
AsymptoticDSolveValue[z^2*y''[z]-3/2*z*y'[z]+(1+z)*y[z]==0,y[z],{z,0,5}]
\[ y(z)\to c_1 \left (-\frac {4 z^5}{675675}+\frac {2 z^4}{10395}-\frac {4 z^3}{945}+\frac {2 z^2}{35}-\frac {2 z}{5}+1\right ) z^2+c_2 \left (\frac {4 z^5}{1575}-\frac {2 z^4}{45}+\frac {4 z^3}{9}-2 z^2+2 z+1\right ) \sqrt {z} \]