Exercise

1. The following differential equations have the solution of the form $e^{mx}$. Find the $n$ independent solutions and the general solution. Finally,show the $n$ solutions are linearly independent.
$$\begin{array}{ll} (a) \ y^{\prime\prime\prime} + y^{\prime\pr... ...\prime\prime} - 8y^{\prime\prime} + 7y^{\prime} = 0 \end{array}$$
2. The following differential equations have the solution of the form either $\cos{mx}$ or $\sin{mx}$. Find the general solution.
$(a) \ y^{\prime\prime} + 4y = 0 \ \ (b) \ y^{(4)} + 4y^{\prime\prime} + 3y = 0$
3. The following differential equations have the solution of the form $x^{m}$. Find the general solution.
$(a) \ x^{2}y^{\prime\prime} + xy^{\prime} - 4y = 0\ \ (b) \ x^{2}y^{\prime\prime} - xy^{\prime} - 3y = 0$