Q21E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

First-Order Constant-Coefficient Equations.

 

1. Substituting y = e^rt, find the auxiliary equation for the first-order linear equation ay'+by = 0, where a and b are constants with $\mathit{a}\mathbf{\ne }\mathbf{0}$.
2. Use the result of part (a) to find the general solution.

Step-by-Step Solution

Answer

The general solution is $y\left(t\right)=c{e}^{-\frac{b}{a}t}$, where c is any constant.

Step 1: Find the auxiliary equation.

The given differential equation is ay' +by = 0.

If y = e^rt then y' = re^rt.

Now substitute the all values in the equation ay'+by = 0, then;

$\begin{array}{rcl}a\left(r{e}^{rt}\right)+b\left(e^{rt}\right)& =& 0\\ (ar+b)e^{rt}& =& 0\\ ar+b& =& 0\end{array}$

Therefore, the auxiliary equation is (ar+b) = 0

Step 2: Find the general solution.

Find the roots of the auxiliary equation.

$$\begin{gathered} (ar+b)=0 \\ r=-\frac{b}{a} \end{gathered}$$

Therefore  $y\left(t\right)=e^{-\frac{b}{a}t}$.

Thus, the general solution is $y\left(t\right)=c{e}^{-\frac{b}{a}t}$, where c is any constant.