\( 2 y ^ { \prime \prime } - 5 y ^ { \prime } - 3 y = 0 \)

This is a second-order linear homogeneous differential equation with constant coefficients. To solve it, you can start by assuming a solution of the form \( y = e^{rt} \), where \( r \) is a constant to be determined. Substituting this assumption into the equation leads you to find the characteristic polynomial \( 2r^2 - 5r - 3 = 0 \). Solving this quadratic equation gives you the roots that will help construct the general solution. After you find the roots \( r_1 \) and \( r_2 \), the general solution will take the form \( y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} \), where \( C_1 \) and \( C_2 \) are constants determined by initial conditions or boundary values, should they be given. Happy solving!