76.12.16 problem 28

Internal problem ID [17572]
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 : Tuesday, January 28, 2025 at 10:44:23 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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 49

dsolve([a*diff(y(t),t$2)+b*diff(y(t),t)+c*y(t)=0,exp(-b*t/(2*a))],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 55

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

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