Compute Symbolic Solutions for ODEs with Events 

Solve an ODE with a state-dependent event using DSolve.

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X sol = DSolve[{y''[t] == -981/100, y[0] == 5, y'[0] == 0, WhenEvent[y[t] == 0, y'[t] -> -(7/10) y'[t]]}, y[t], {t, 0, 5}]

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This system models a bouncing ball.

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X Plot[y[t] /. sol, {t, 0, 5}, Filling -> Axis]

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Stop the integration of an ODE when an event occurs.

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X DSolve[{x'[t] == -x[t], x[0] == 1, WhenEvent[t == 3, "StopIntegration"]}, x, {t, 0, 4}]

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Remove an event after it has occurred once.

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X sol = DSolve[{x''[t] + x[t] == 0, x[0] == 0, x'[0] == 1, WhenEvent[x[t] == 0, {x'[t] -> -x'[t], "RemoveEvent"}]}, x, {t, 0, 10}]

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X Plot[x[t] /. sol, {t, 0, 10}]

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Specify that a variable maintains its value between events.

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X sol = DSolve[{x'[t] == a[t], x[0] == 1, a[0] == 1, WhenEvent[Mod[x[t], 1] == 0, a[t] -> - a[t]]}, {x, a}, {t, 0, 5}, DiscreteVariables -> a]

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X Plot[Evaluate[{x[t], a[t]} /. sol], {t, 0, 4}, Exclusions -> None]

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