Chapter02 - Home

Exercise. This is the solution to exercise 2.1 in the book.

Solution. The characteristic equation is:

$${m}^{2}+m-6=0$$with solutions

$${m}_{1}=2\phantom{\rule{1em}{0ex}}{m}_{2}=-3$$Hence, the general solution is:

$$y\left(x\right)={c}_{1}{e}^{2x}+{c}_{2}{e}^{-3x}$$For

$$\begin{array}{rcll}y\left(0\right)& =& 0& \text{}\\ {y}^{\prime}\left(0\right)& =& 1& \text{}\end{array}$$or

$$\begin{array}{rcll}0& =& {c}_{1}+{c}_{2}& \text{}\\ 1& =& 2{c}_{1}-3{c}_{2}& \text{}\end{array}$$Hence:

$$y\left(x\right)=\frac{1}{5}{e}^{2x}-\frac{1}{5}{e}^{-3x}$$