90.20.31 problem 18 (3 d)

Internal problem ID [25326]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 337
Problem number : 18 (3 d)
Date solved : Friday, October 03, 2025 at 12:00:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }-4 t y^{\prime }+6 y&=t^{5} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=a \\ y^{\prime }\left (1\right )&=b \\ \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 36

ode:=t^2*diff(diff(y(t),t),t)-4*t*diff(y(t),t)+6*y(t) = t^5; 
ic:=[y(1) = a, D(y)(1) = b]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {t^{5}}{6}+\frac {\left (2 b -1-4 a \right ) t^{3}}{2}+\frac {\left (9 a -3 b +1\right ) t^{2}}{3} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 34

ode=t^2*D[y[t],{t,2}]-4*t*D[y[t],t]+6*y[t]==t^5; 
ic={y[1]==a,Derivative[1][y][1] ==b}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{6} t^2 \left ((t-1) \left (6 b+t^2+t-2\right )-6 a (2 t-3)\right ) \end{align*}

✓ Sympy. Time used: 0.227 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-t**5 + t**2*Derivative(y(t), (t, 2)) - 4*t*Derivative(y(t), t) + 6*y(t),0) 
ics = {y(1): a, Subs(Derivative(y(t), t), t, 1): b} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t^{2} \left (3 a - b + \frac {t^{3}}{6} + t \left (- 2 a + b - \frac {1}{2}\right ) + \frac {1}{3}\right ) \]