Problem 77 - University Calculus: Early Transcendentals

Question

Solve the initial value problems $$\frac{d^{2} s}{d t^{2}}=-4 \sin \left(2 t-\frac{\pi}{2}\right), \quad s^{\prime}(0)=100, \quad s(0)=0$$

Step-by-Step Solution

Answer

The solution is \( s(t) = \sin(2t - \frac{\pi}{2}) + 100t \).

Step 1: Integrate the Second Derivative

We start by integrating the second derivative to find the first derivative. Given \( \frac{d^2 s}{d t^2} = -4 \sin \left( 2t - \frac{\pi}{2} \right) \), we integrate with respect to \( t \). The integral of \( \sin(2t - \frac{\pi}{2}) \) is \( -\frac{1}{2}\cos(2t - \frac{\pi}{2}) \). Thus, the first derivative is \( s'(t) = 2\cos(2t - \frac{\pi}{2}) + C_1 \), where \( C_1 \) is a constant of integration.

Step 2: Apply Initial Condition for First Derivative

Use the initial condition \( s'(0) = 100 \) to find \( C_1 \). Plug \( t = 0 \) into the first derivative equation: \( s'(0) = 2\cos\left(-\frac{\pi}{2}\right) + C_1 = 100 \). Since \( \cos(-\frac{\pi}{2}) = 0 \), we have \( C_1 = 100 \). Thus, the equation becomes \( s'(t) = 2\cos(2t - \frac{\pi}{2}) + 100 \).

Step 3: Integrate the First Derivative

Now integrate \( s'(t) = 2\cos(2t - \frac{\pi}{2}) + 100 \) to find \( s(t) \). The integral of \( 2\cos(2t - \frac{\pi}{2}) \) is \( \sin(2t - \frac{\pi}{2}) \) as \( \frac{d}{dt} \sin(2t - \frac{\pi}{2}) \) is \( 2\cos(2t - \frac{\pi}{2}) \). Thus, \( s(t) = \sin(2t - \frac{\pi}{2}) + 100t + C_2 \), where \( C_2 \) is another constant of integration.

Step 4: Apply Initial Condition for Initial Value

Use the initial condition \( s(0) = 0 \) to find \( C_2 \). Plug \( t = 0 \) into \( s(t) = \sin(2t - \frac{\pi}{2}) + 100t + C_2 \): \( s(0) = \sin(0) + 0 + C_2 = 0 \). Therefore, \( C_2 = 0 \). Thus, \( s(t) = \sin(2t - \frac{\pi}{2}) + 100t \).

Key Concepts

Initial Value Problems

Initial value problems (IVPs) involve finding a particular solution to a differential equation that satisfies given initial conditions. These initial conditions typically specify the value of the function and possibly its derivatives at a certain point.
Consider a general second-order differential equation and initial conditions:

• Differential equation: \( \frac{d^{2} s}{dt^{2}} = f(t) \)
• Initial conditions: Values are given for the function \( s(t) \) and its first derivative \( s'(t) \) at a particular time \( t = t_0 \).

The solution is determined by integrating the differential equation and using the initial conditions to find the integration constants. This ensures that the solution is not just any solution to the differential equation but the specific one that meets the criteria given by the initial values.

Second Order Differential Equations

Second order differential equations involve the second derivative of a function.
In our example, we have the equation \( \frac{d^{2}s}{dt^{2}} = -4 \sin(2t - \frac{\pi}{2}) \). This suggests that the second derivative of \( s(t) \) with respect to \( t \) is given in terms of trigonometric functions.
The solution process involves finding the first derivative through integration, and then solving for \( s(t) \), the original function. Each integration introduces an unknown constant, which needs to be determined using initial conditions.
Initial conditions are crucial here for calculating these constants, thereby yielding a unique solution that fits the initial setup. In practice, second order differential equations appear in various physics problems like oscillations and wave dynamics.

Integration Techniques

Integration is a key technique when solving differential equations. For an equation like \( \frac{d^{2}s}{dt^{2}} = -4 \sin(2t - \frac{\pi}{2}) \), we employ integration to reverse differentiation.
Here are the general steps for solving the example problem:

• First Integration: Integrating \( -4 \sin(2t - \frac{\pi}{2}) \) with respect to \( t \) to find the first derivative \( s'(t) \). This gives us \( 2\cos(2t - \frac{\pi}{2}) + C_1 \).
• Determine the Constant \( C_1 \): Apply the initial condition \( s'(0) = 100 \) to find \( C_1 \).
• Second Integration: Further integrate the first derivative to find the function \( s(t) \). This results in \( \sin(2t - \frac{\pi}{2}) + 100t + C_2 \).
• Determine Constant \( C_2 \): Use the initial condition \( s(0) = 0 \) to find \( C_2 \).

Through this process, integration allows us to reconstruct the original function from its derivatives, a crucial step in solving differential equations.