Question

onsider the initial value problem y′′+7y′+12y=0, y(0)=6, y′(0)=β, where β>0. (b) Determine the coordinates tm and ymof the maximum point of the solution as functions of β. Enclose arguments of functions in parentheses. For example, sin(2x).

Answer 1

To find the coordinates of the maximum point of the solution, solve the differential equation and differentiate the solution to find the time when velocity is zero (tm). Substituting tm into the solution will give the corresponding ym.

Step-by-step explanation:

The coordinates tm and ym of the maximum point of the solution can be determined by finding the time (tm) when the velocity (dy/dt) of the solution is zero.

To do this, we need to solve the differential equation y′′+7y′+12y=0 and find the solution y(t) for different values of β. Then, we can differentiate y(t) with respect to t and set it equal to zero to find the value of tm.

Finally, substitute tm into y(t) to find the corresponding ym.