56.1.48 problem 48

Internal problem ID [8760]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 48
Date solved : Sunday, March 30, 2025 at 01:30:36 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} \left (t^{2}+9\right ) y^{\prime \prime }+2 t y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (3\right )&=2 \pi \\ y^{\prime }\left (3\right )&={\frac {2}{3}} \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 12

ode:=(t^2+9)*diff(diff(y(t),t),t)+2*t*diff(y(t),t) = 0; 
ic:=y(3) = 2*Pi, D(y)(3) = 2/3; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \pi +4 \arctan \left (\frac {t}{3}\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 15

ode=(t^2+9)*D[y[t],{t,2}]+2*t*D[y[t],t]==0; 
ic={y[3]==2*Pi,Derivative[1][y][3 ]==2/3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 4 \arctan \left (\frac {t}{3}\right )+\pi \]

✓ Sympy. Time used: 0.241 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*t*Derivative(y(t), t) + (t**2 + 9)*Derivative(y(t), (t, 2)),0) 
ics = {y(3): 2*pi, Subs(Derivative(y(t), t), t, 3): 2/3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 4 \operatorname {atan}{\left (\frac {t}{3} \right )} + \pi \]