Problem 108 - University Calculus: Early Transcendentals

Question

Solve the initial value problems in Exercises. $$\frac{d^{2} s}{d t^{2}}=\frac{3 t}{8} ;\left.\quad \frac{d s}{d t}\right|_{t=4}=3, \quad s(4)=4$$

Step-by-Step Solution

Answer

The solution is \( s(t) = \frac{1}{16} t^3 \).

Step 1: Identify the Problem

The problem involves solving a second-order differential equation with given initial conditions. The differential equation is \( \frac{d^{2} s}{d t^{2}} = \frac{3 t}{8} \), with initial conditions \( \left.\frac{d s}{d t}\right|_{t=4}=3 \) and \( s(4)=4 \).

Step 2: Integrate the Second Order Differential Equation

To solve the differential equation, we first integrate \( \frac{d^{2}s}{dt^{2}} = \frac{3t}{8} \) with respect to \( t \) to find \( \frac{ds}{dt} \). Integration yields: \[ \frac{d s}{d t} = \int \frac{3 t}{8} dt = \frac{3}{16} t^2 + C_1 \].

Step 3: Apply Initial Condition for First Derivative

Use the initial condition \( \left.\frac{d s}{d t}\right|_{t=4}=3 \) to find \( C_1 \). Substitute \( t = 4 \) and \( \frac{ds}{dt} = 3 \), giving:\[ 3 = \frac{3}{16} (4)^2 + C_1 \]. Solving for \( C_1 \), we find: \[ 3 = 3 + C_1 \] \( C_1 = 0 \).

Step 4: Integrate to Find the Original Function

Now, integrate \( \frac{d s}{d t} = \frac{3}{16} t^2 \) to find \( s(t) \). This gives us:\[ s(t) = \int \frac{3}{16} t^2 dt = \frac{3}{48} t^3 + C_2 \].Simplifying, \[ s(t) = \frac{1}{16} t^3 + C_2 \].

Step 5: Apply Initial Condition for the Original Function

Apply the second initial condition \( s(4) = 4 \) to find \( C_2 \). Substitute \( t = 4 \) into the equation \( s(t) = \frac{1}{16} t^3 + C_2 \):\[ 4 = \frac{1}{16} (4)^3 + C_2 \]. Solving for \( C_2 \), we find:\[ 4 = 4 + C_2 \] \( C_2 = 0 \).

Step 6: Write the Solution

With both constants found, the solution to the initial value problem is:\[ s(t) = \frac{1}{16} t^3 \].

Key Concepts

Initial Value Problems

Initial value problems (IVPs) are a fundamental concept in the study of differential equations. In these problems, we are tasked with finding a function that satisfies a given differential equation and also conforms to specified starting conditions, known as initial conditions. These conditions are typically given in the form of specific values of the function and its derivatives at a particular point in time.

When solving an initial value problem, we follow these crucial steps:

• Identify the differential equation and the initial conditions provided.
• Solve the differential equation, usually by integration.
• Apply the initial conditions to determine any unknown constants in the general solution.

In the original exercise, we are given a second-order differential equation along with two initial conditions. The solution process involves integrating the equation and using the initial conditions to find the specific solution that fits the problem's requirements.

Second-Order Differential Equation

A second-order differential equation involves the second derivative of a function. The notation used is typically in the form \( \frac{d^2y}{dx^2} \) or, as in our problem, \( \frac{d^{2} s}{d t^{2}} \). These equations are frequently encountered in physics and engineering, as they often describe systems with acceleration, such as motion under gravity.

The process of solving a second-order differential equation typically involves:

• Finding the first derivative of the function by integrating the second derivative.
• Finding the original function by further integrating the first derivative.
• Applying initial conditions to find any constants introduced during integration.

In the exercise, we take the given second-order equation \( \frac{d^{2} s}{d t^{2}} = \frac{3 t}{8} \) and integrate it twice, once to find the first derivative and then to derive the function \( s(t) \). The initial conditions are essential in pinning down exact values for the constants of integration.

Integration Techniques

Integration is a core technique used in solving differential equations. It involves finding a function whose derivative is the integrand. This step is essential in transforming a derivative equation into a solvable function.

For our second-order problem, integration techniques were used as follows:

• First, integrate \( \frac{d^{2} s}{d t^{2}} = \frac{3 t}{8} \) to find \( \frac{d s}{d t} \). This yields a function in terms of \( t \) with an integration constant \( C_1 \).
• Apply the given initial condition \( \left.\frac{d s}{d t}\right|_{t=4}=3 \) to find \( C_1 \).
• Next, integrate \( \frac{d s}{d t} = \frac{3}{16} t^2 + C_1 \) to get the function \( s(t) \), introducing another constant \( C_2 \).
• Use the initial condition \( s(4)=4 \) to solve for \( C_2 \).

These integration techniques help transition from solving the differential form to obtaining an explicit solution that aligns with the given initial conditions.