74.18.46 problem 53

Internal problem ID [16524]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 53
Date solved : Thursday, March 13, 2025 at 08:16:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 t y^{\prime }+t^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-2*t*diff(y(t),t)+t^2*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{\frac {t^{2}}{2}} \left (c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} \right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 39

ode=D[y[t],{t,2}]-2*t*D[y[t],t]+t^2*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{2} e^{\frac {1}{2} t (t-2 i)} \left (2 c_1-i c_2 e^{2 i t}\right ) \]

✓ Sympy. Time used: 0.816 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*y(t) - 2*t*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {3 t^{5} r{\left (3 \right )}}{10} + C_{2} \left (- \frac {t^{6}}{45} - \frac {t^{4}}{12} + 1\right ) + C_{1} t \left (1 - \frac {t^{4}}{20}\right ) + O\left (t^{6}\right ) \]