Internal problem ID [2033]
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 16.14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
\[ \boxed {z y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y=0} \] With the expansion point for the power series method at \(z = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 309
Order:=6; dsolve(z*diff(y(z),z$2)+(1-z)*diff(y(z),z)+lambda*y(z)=0,y(z),type='series',z=0);
\[ y \left (z \right ) = \left (\left (2 \lambda +1\right ) z +\left (\frac {1}{4} \lambda +\frac {1}{4}-\frac {3}{4} \lambda ^{2}\right ) z^{2}+\left (-\frac {2}{9} \lambda ^{2}+\frac {1}{27} \lambda +\frac {1}{18}+\frac {11}{108} \lambda ^{3}\right ) z^{3}+\left (\frac {7}{192} \lambda ^{3}-\frac {167}{3456} \lambda ^{2}+\frac {1}{192} \lambda +\frac {1}{96}-\frac {25}{3456} \lambda ^{4}\right ) z^{4}+\left (\frac {1}{1500} \lambda -\frac {37}{4320} \lambda ^{2}+\frac {719}{86400} \lambda ^{3}-\frac {61}{21600} \lambda ^{4}+\frac {137}{432000} \lambda ^{5}+\frac {1}{600}\right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) c_{2} +\left (1-\lambda z +\frac {1}{4} \left (-1+\lambda \right ) \lambda z^{2}-\frac {1}{36} \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{3}+\frac {1}{576} \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{4}-\frac {1}{14400} \left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \left (c_{2} \ln \left (z \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 415
AsymptoticDSolveValue[z*y''[z]+(1-z)*y'[z]+\[Lambda]*y[z]==0,y[z],{z,0,5}]
\[ y(z)\to c_1 \left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda z^3+\frac {1}{4} (\lambda -1) \lambda z^2-\lambda z+1\right )+c_2 \left (\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) z^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -2) \lambda z^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -1) \lambda z^5}{14400}+\frac {(\lambda -4) (\lambda -2) (\lambda -1) \lambda z^5}{14400}+\frac {137 (\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{432000}+\frac {(\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{14400}-\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) z^4-\frac {1}{576} (\lambda -3) (\lambda -2) \lambda z^4-\frac {1}{576} (\lambda -3) (\lambda -1) \lambda z^4-\frac {25 (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^4}{3456}-\frac {1}{576} (\lambda -2) (\lambda -1) \lambda z^4+\frac {1}{36} (\lambda -2) (\lambda -1) z^3+\frac {1}{36} (\lambda -2) \lambda z^3+\frac {11}{108} (\lambda -2) (\lambda -1) \lambda z^3+\frac {1}{36} (\lambda -1) \lambda z^3-\frac {1}{4} (\lambda -1) z^2-\frac {3}{4} (\lambda -1) \lambda z^2-\frac {\lambda z^2}{4}+\left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda z^3+\frac {1}{4} (\lambda -1) \lambda z^2-\lambda z+1\right ) \log (z)+2 \lambda z+z\right ) \]