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First Order Differential Equations

These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

dy/dx = f(x)

If function \(f(x)\) satisfies \[\frac{df(x)}{dx}=(3x+4)(2-x) \text{ and } f(-1)=18, \] what is \(f(x)?\)

If \(f'(x)=9x^2\) and \(f(2)=31,\) what is \(f(x)?\)

If \[f'(x)=\sin x-2\cos x, f(\pi)=9,\] what is \(f(x)?\)

Suppose that the acceleration \(a(t)\) of a particle at time \(t,\) is given by the formula \(a(t)=4t-3.\) If \[v(1)=8, f(2)=5,\] where \(v(t)\) and \(f(t)\) are the velocity and position function of the particle, respectively, what is the position of the particle when \(t=1 ?\)

The velocity at time \(t\) of a particle moving along the \(x\)-axis is given by the formula \[v(t)=8t^3-9t^2.\] If the initial position of the particle on the \(x\)-axis is \(x=9\), what is its position when \(t=2 ?\)

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