Q22E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 22–25, use the method described in Problem 21to find a general solution to the given equation.

22. 3Y' - 7Y = 0

Step-by-Step Solution

Answer

The general solution is $y\left(t\right)=c{e}^{\frac{7}{3}t}$.

Step 1: Substitute y = e rt

The given differential equation is 3y' - 7y = 0

Substitutey = e^rt and y' = re^rt,

$\begin{array}{rcl}3\left(r{e}^{rt}\right)-7\left(e^{rt}\right)& =& 0\\ (3r-7)\left(e^{rt}\right)& =& 0\\ 3r-7& =& 0\\ r& =& \frac{7}{3}\end{array}$

Step 2: Find the general solution.

Therefore $y\left(t\right)=e^{\frac{7}{3}t}$.

Thus, the general solution is $y\left(t\right)=c{e}^{\frac{7}{3}t}$.