Internal problem ID [2028]
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 16.10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {z \left (1-z \right ) y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y=0} \end {gather*} With the expansion point for the power series method at \(z = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 261
Order:=6; dsolve(z*(1-z)*diff(y(z),z$2)+(1-z)*diff(y(z),z)+lambda*y(z)=0,y(z),type='series',z=0);
\[ y \relax (z ) = \left (2 \lambda z +\left (\frac {1}{4} \lambda -\frac {3}{4} \lambda ^{2}\right ) z^{2}+\left (-\frac {37}{108} \lambda ^{2}+\frac {2}{27} \lambda +\frac {11}{108} \lambda ^{3}\right ) z^{3}+\left (\frac {139}{1728} \lambda ^{3}-\frac {649}{3456} \lambda ^{2}+\frac {1}{32} \lambda -\frac {25}{3456} \lambda ^{4}\right ) z^{4}+\left (-\frac {13}{1600} \lambda ^{4}+\frac {8467}{144000} \lambda ^{3}-\frac {2527}{21600} \lambda ^{2}+\frac {2}{125} \lambda +\frac {137}{432000} \lambda ^{5}\right ) z^{5}+\mathrm {O}\left (z^{6}\right )\right ) c_{2}+\left (1-\lambda z +\frac {1}{4} \left (-1+\lambda \right ) \lambda z^{2}-\frac {1}{36} \lambda \left (\lambda ^{2}-5 \lambda +4\right ) z^{3}+\frac {1}{576} \lambda \left (\lambda ^{3}-14 \lambda ^{2}+49 \lambda -36\right ) z^{4}-\frac {1}{14400} \lambda \left (-1+\lambda \right ) \left (\lambda -4\right ) \left (\lambda -16\right ) \left (\lambda -9\right ) z^{5}+\mathrm {O}\left (z^{6}\right )\right ) \left (c_{2} \ln \relax (z )+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 940
AsymptoticDSolveValue[z*(1-z)*y''[z]+(1-z)*y'[z]+\[Lambda]*y[z]==0,y[z],{z,0,5}]
\[ y(z)\to \left (\frac {1}{25} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{16} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) z^5+\frac {1}{16} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) z^4+\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) z^3+\frac {1}{4} \left (\lambda ^2-\lambda \right ) z^2-\lambda z+1\right ) c_1+c_2 \left (-\frac {2}{125} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{16} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) z^5+\frac {1}{25} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda +\frac {2}{27} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{9} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda \right ) \lambda +\frac {1}{32} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{16} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda +\frac {2}{27} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{9} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda \right ) \lambda \right ) \lambda \right ) z^5-\frac {1}{32} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) z^4+\frac {1}{16} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda +\frac {2}{27} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{9} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda \right ) \lambda \right ) z^4-\frac {2}{27} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) z^3+\frac {1}{9} \left (\frac {\lambda ^3}{2}-2 \lambda ^2+\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda \right ) z^3-\frac {\lambda ^2 z^2}{2}-\frac {1}{4} \left (\lambda ^2-\lambda \right ) z^2+2 \lambda z+\left (\frac {1}{25} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\frac {1}{16} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) z^5+\frac {1}{16} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) \lambda -\lambda \right ) z^4+\frac {1}{9} \left (\lambda ^2-\frac {1}{4} \left (\lambda ^2-\lambda \right ) \lambda -\lambda \right ) z^3+\frac {1}{4} \left (\lambda ^2-\lambda \right ) z^2-\lambda z+1\right ) \log (z)\right ) \]