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## Homework Statement

{Prob #27, Section 2.1 "EDE" (Boyce/Prima, 10thEd), pp. 40}

"Consider the initial value problem [and] find the coordinates for the first local maximum point of the solution, t>0."

## Homework Equations

y' + (1/2)y = 2cost y(0) = -1

## The Attempt at a Solution

I began this thinking it would be fairly straightforward, and readily found the general solution:

y = (4/5)(cost + 2sint) + ce^(-t/2)

To find "c" (ie: the specific solution) I used the initial values and got:

y=(4/5)(cost + 2sint) - (9/5)e^(-t/2)

In attempting to find the (first) local maxima -- making the assumption that it would be at a point where y' = 0 -- I took the first equation, which reduces to:

(1/2)y = 2cost ----> y = 4cost

and tried solving for t [by inserting my above-noted specific equation for y(t)]. Which ultimately brings me to the equation:

9e^(-t/2) = 8sint - 16cost

Now I should note that [since the answer, meaning the point value (1.364312, 0.820082) is in the back of the book] all my equations balance. So I assume I've done the various derivations and integrals correctly. My general and specific solutions seem correct. But I cannot for the LIFE of me think of how to simplify the equation above to some form where "t" can be solve for. (Though as I mentioned, putting the 'known' value in does balance.)

Logically I'm overlooking something that should either allow me to simplify that equation, or I've created the wrong equation. But I'm stuck.

[FWIW - I'm not formally taking a class; I'm considering a project that requires re-building of my DiffyQ knowledge, now 30ish years stale. So I recently worked through the most current version of Thomas' Calculus and am now working through this (introductory) ODE text. If I had an instructor or TA grad student, I'd be bugging one of them. But I live in a town sans 4-year college, so I can't think who to 'bug'! Thanks for considering help!]