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copied!<p>Use <strong>NMaximize</strong></p> <p>First approximation: </p> <pre><code>s = NDSolve[{x''[t] + x[t] - 0.167 x[t]^3 == 0.005 Cos[t + -0.0000977162*t^2/2], x[0] == 0, x'[0] == 0}, x[t], {t, 0, 1000}] NMaximize[{Evaluate[x[t] /. s[[1]]] , 100 < t < 1000}, t] {1.26625, {t -> 821.674}} </code></pre> <p>As your function is a rapid oscillation like this : <img src="https://i.stack.imgur.com/SJ4XU.png" alt="alt text"> , it doesn't catch the real max value, as you may see below: </p> <pre><code>Plot[{1.26625, Evaluate[x[t] /. s[[1]]]}, {t, 790, 830}, Frame -> {True, True, False, False}, FrameLabel -> {"t", "x"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False, PlotRange -> {{790, 830}, {1.25, 1.27}}] </code></pre> <p><img src="https://i.stack.imgur.com/g0y45.png" alt="alt text"></p> <p>So we refine our bounds, and tune a little the NMaximize function: </p> <pre><code>NMaximize[{Evaluate[x[t] /. s[[1]]] , 814 < t < 816}, t, AccuracyGoal -> 20, PrecisionGoal -> 18, MaxIterations -> 1000] NMaximize::cvmit: Failed to converge to the requested accuracy or precision within 1000 iterations. >> {1.26753, {t -> 814.653}} </code></pre> <p>It failed to converge within the required accuracy, but now the result is good enough</p> <pre><code>Plot[{1.2675307922753962`, Evaluate[x[t] /. s[[1]]]}, {t, 790, 830}, Frame -> {True, True, False, False}, FrameLabel -> {"t", "x"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False, PlotRange -> {{790, 830}, {1.25, 1.27}}] </code></pre> <p><img src="https://i.stack.imgur.com/8O0ax.png" alt="alt text"></p>

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