90.31.9 problem 9
Internal
problem
ID
[25459]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
7.
Power
series
methods.
Exercises
at
page
537
Problem
number
:
9
Date
solved
:
Friday, October 03, 2025 at 12:01:44 AM
CAS
classification
:
[_Laguerre]
\begin{align*} t y^{\prime \prime }+\left (1-t \right ) y^{\prime }+\lambda y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 154
Order:=6;
ode:=t*diff(diff(y(t),t),t)+(1-t)*diff(y(t),t)+lambda*y(t) = 0;
dsolve(ode,y(t),type='series',t=0);
\[
y = \left (\left (2 \lambda +1\right ) t +\left (\frac {1}{4} \lambda +\frac {1}{4}-\frac {3}{4} \lambda ^{2}\right ) t^{2}+\left (-\frac {2}{9} \lambda ^{2}+\frac {1}{27} \lambda +\frac {1}{18}+\frac {11}{108} \lambda ^{3}\right ) t^{3}+\left (\frac {7}{192} \lambda ^{3}-\frac {167}{3456} \lambda ^{2}+\frac {1}{192} \lambda +\frac {1}{96}-\frac {25}{3456} \lambda ^{4}\right ) t^{4}+\left (-\frac {61}{21600} \lambda ^{4}+\frac {137}{432000} \lambda ^{5}+\frac {1}{600}+\frac {719}{86400} \lambda ^{3}-\frac {37}{4320} \lambda ^{2}+\frac {1}{1500} \lambda \right ) t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 +\left (1-\lambda t +\frac {1}{4} \left (-1+\lambda \right ) \lambda t^{2}-\frac {1}{36} \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda t^{3}+\frac {1}{576} \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda t^{4}-\frac {1}{14400} \left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \left (c_2 \ln \left (t \right )+c_1 \right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 415
ode=t*D[y[t],{t,2}]+(1-t)*D[y[t],t]+\[Lambda]*y[t]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
\[
y(t)\to c_1 \left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda t^3+\frac {1}{4} (\lambda -1) \lambda t^2-\lambda t+1\right )+c_2 \left (\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) t^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -2) \lambda t^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -1) \lambda t^5}{14400}+\frac {(\lambda -4) (\lambda -2) (\lambda -1) \lambda t^5}{14400}+\frac {137 (\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{432000}+\frac {(\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{14400}-\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) t^4-\frac {1}{576} (\lambda -3) (\lambda -2) \lambda t^4-\frac {1}{576} (\lambda -3) (\lambda -1) \lambda t^4-\frac {25 (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^4}{3456}-\frac {1}{576} (\lambda -2) (\lambda -1) \lambda t^4+\frac {1}{36} (\lambda -2) (\lambda -1) t^3+\frac {1}{36} (\lambda -2) \lambda t^3+\frac {11}{108} (\lambda -2) (\lambda -1) \lambda t^3+\frac {1}{36} (\lambda -1) \lambda t^3-\frac {1}{4} (\lambda -1) t^2-\frac {3}{4} (\lambda -1) \lambda t^2-\frac {\lambda t^2}{4}+\left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda t^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda t^3+\frac {1}{4} (\lambda -1) \lambda t^2-\lambda t+1\right ) \log (t)+2 \lambda t+t\right )
\]
✓ Sympy. Time used: 0.354 (sec). Leaf size: 73
from sympy import *
t = symbols("t")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*y(t) + t*Derivative(y(t), (t, 2)) + (1 - t)*Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
\[
y{\left (t \right )} = C_{1} \left (- \frac {\lambda _{} t^{5} \left (\lambda _{} - 4\right ) \left (\lambda _{} - 3\right ) \left (\lambda _{} - 2\right ) \left (\lambda _{} - 1\right )}{14400} + \frac {\lambda _{} t^{4} \left (\lambda _{} - 3\right ) \left (\lambda _{} - 2\right ) \left (\lambda _{} - 1\right )}{576} - \frac {\lambda _{} t^{3} \left (\lambda _{} - 2\right ) \left (\lambda _{} - 1\right )}{36} + \frac {\lambda _{} t^{2} \left (\lambda _{} - 1\right )}{4} - \lambda _{} t + 1\right ) + O\left (t^{6}\right )
\]