90.31.5 problem 5

Internal problem ID [25455]
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 : 5
Date solved : Friday, October 03, 2025 at 12:01:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+\left (1-t \right ) y^{\prime }+4 y t&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 42

Order:=6; 
ode:=t*diff(diff(y(t),t),t)+(1-t)*diff(y(t),t)+4*t*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-t^{2}-\frac {2}{9} t^{3}+\frac {5}{24} t^{4}+\frac {31}{450} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (t +\frac {5}{4} t^{2}-\frac {7}{54} t^{3}-\frac {131}{288} t^{4}-\frac {481}{6750} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 99

ode=t*D[y[t],{t,2}]+(1-t)*D[y[t],t]+4*t*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {31 t^5}{450}+\frac {5 t^4}{24}-\frac {2 t^3}{9}-t^2+1\right )+c_2 \left (-\frac {481 t^5}{6750}-\frac {131 t^4}{288}-\frac {7 t^3}{54}+\frac {5 t^2}{4}+\left (\frac {31 t^5}{450}+\frac {5 t^4}{24}-\frac {2 t^3}{9}-t^2+1\right ) \log (t)+t\right ) \]

✓ Sympy. Time used: 0.302 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*t*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)
 

\[ y{\left (t \right )} = C_{1} \left (\frac {31 t^{5}}{450} + \frac {5 t^{4}}{24} - \frac {2 t^{3}}{9} - t^{2} + 1\right ) + O\left (t^{6}\right ) \]