Initial Value Problems
Solve the initial value problems in Exercises 89–92.
d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

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Identify the given differential equation and initial conditions: \(\frac{d^3 r}{dt^3} = -\cos t\), with \(r''(0) = 0\), \(r'(0) = 0\), and \(r(0) = -1\).

Integrate the third derivative \(\frac{d^3 r}{dt^3} = -\cos t\) once with respect to \(t\) to find the second derivative \(r''(t)\). Remember to add an integration constant \(C_1\).

Integrate \(r''(t)\) to find the first derivative \(r'(t)\), adding another integration constant \(C_2\).

Integrate \(r'(t)\) to find the original function \(r(t)\), adding a third integration constant \(C_3\).

Use the initial conditions \(r''(0) = 0\), \(r'(0) = 0\), and \(r(0) = -1\) to set up equations and solve for the constants \(C_1\), \(C_2\), and \(C_3\).

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Higher-order differential equations involve derivatives of order two or more. In this problem, the third derivative of r with respect to t is given, requiring integration multiple times to find the original function r(t). Understanding how to reduce the order by successive integration is essential.

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Solving the differential equation requires integrating the function -cos(t) multiple times. Familiarity with the integrals of sine and cosine functions, including their signs and constants of integration, is crucial to correctly find r(t) from its third derivative.

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Domanda del libro di testo

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

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Domanda del libro di testo

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