4.9.16 problem 16

Internal problem ID [1318]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.4 Repeated roots, reduction of order , page 172
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 04:32:34 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-y^{\prime }+\frac {y}{4}&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)-diff(y(t),t)+1/4*y(t) = 0; 
ic:=[y(0) = 2, D(y)(0) = b]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = {\mathrm e}^{\frac {t}{2}} \left (2+\left (b -1\right ) t \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 20

ode=D[y[t],{t,2}]-D[y[t],t]+25/100*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] ==b}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{t/2} ((b-1) t+2) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 14

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

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