Q25E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

25. 6w' - 13w = 0

Step-by-Step Solution

Answer

The general solution is $\mathrm{w}\left(\mathrm{t}\right)={\mathrm{ce}}^{\frac{13}{6}\mathrm{t}}$.

Step 1: Substitute w = e rt

The first order equation is 6w' - 13w = 0

Substitute w = e^rt and w' = er^rt,

$\begin{array}{rcl}6\left(r{e}^{rt}\right)-13\left(e^{rt}\right)& =& 0\\ (6r-13)\left(e^{rt}\right)& =& 0\end{array}$

Since e^rt can’t be zero. Therefore,

$r=\frac{13}{6}$

Step 2: Find the general solution.

Therefore, the general solution is $w\left(t\right)=c{e}^{\frac{13}{6}t}$.