76.12.20 problem 32

Internal problem ID [17505]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 32
Date solved : Monday, March 31, 2025 at 04:16:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=t \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 12

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

\[ y = t \left ({\mathrm e}^{t} c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 16

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

\[ y(t)\to t \left (c_2 e^t+c_1\right ) \]

✓ Sympy. Time used: 0.822 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{2} t^{2} \left (\frac {t^{3}}{24} + \frac {t^{2}}{6} + \frac {t}{2} + 1\right ) + C_{1} t + O\left (t^{6}\right ) \]