70.12.16 problem 28

Internal problem ID [18859]
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 : 28
Date solved : Friday, October 03, 2025 at 07:33:55 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} a y^{\prime \prime }+b y^{\prime }+c y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-\frac {b t}{2 a}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 44

ode:=a*diff(diff(y(t),t),t)+b*diff(y(t),t)+c*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (c_1 \,{\mathrm e}^{\frac {t \sqrt {-4 c a +b^{2}}}{a}}+c_2 \right ) {\mathrm e}^{-\frac {\left (b +\sqrt {-4 c a +b^{2}}\right ) t}{2 a}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 55

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

\begin{align*} y(t)&\to e^{-\frac {t \left (\sqrt {b^2-4 a c}+b\right )}{2 a}} \left (c_2 e^{\frac {t \sqrt {b^2-4 a c}}{a}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.149 (sec). Leaf size: 46

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

\[ y{\left (t \right )} = C_{1} e^{\frac {t \left (- b + \sqrt {- 4 a c + b^{2}}\right )}{2 a}} + C_{2} e^{- \frac {t \left (b + \sqrt {- 4 a c + b^{2}}\right )}{2 a}} \]