# Quantum Chemistry

Chapter02 - Home
Exercises: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - 17 - 18 - 19 - 20 - 21 - 22 - 23 - 24 - 25 - 26 - 27 - 28 - 29

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:

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