Problem 66 - Essential Calculus Early Transcendentals

Question

Find \(\mathbf{r}(t)\) if \(\mathbf{r}^{\prime}(t)=t \mathbf{i}+e^{t} \mathbf{j}+t e^{t} \mathbf{k}\) and \(\mathbf{r}(0)=\mathbf{i}+\mathbf{j}+\mathbf{k}\)

Step-by-Step Solution

Answer

\( \mathbf{r}(t) = \left( \frac{t^2}{2} + 1 \right) \mathbf{i} + e^t \mathbf{j} + \left( t e^t - e^t + 2 \right) \mathbf{k} \)

Step 1: Integrate the i-component

The derivative with respect to **t** for the **i-component** of \( \mathbf{r}''(t) \) is **t**. To find the component for \( \mathbf{r}(t) \), we integrate \( t \) with respect to **t**, obtaining \( \frac{t^2}{2} + C_1 \), where \( C_1 \) is the integration constant.

Step 2: Integrate the j-component

The derivative for the **j-component** is \( e^t \). Integrate \( e^t \) with respect to **t**, obtaining \( e^t + C_2 \), where \( C_2 \) is the integration constant.

Step 3: Integrate the k-component

For the **k-component**, \( t e^t \), we integrate by parts: let \( u = t \), \( dv = e^t dt \). Then \( du = dt \) and \( v = e^t \). Using \( \int u \, dv = uv - \int v \, du \): \( t e^t - \int e^t dt = t e^t - e^t + C_3 \), where \( C_3 \) is the integration constant.

Step 4: Determine the Constants

Utilize the known condition \( \mathbf{r}(0) = \mathbf{i} + \mathbf{j} + \mathbf{k} \). Set **t = 0** in the integrations:- \( \frac{0^2}{2} + C_1 = 1 \), so \( C_1 = 1 \).- \( e^0 + C_2 = 1 \), so \( C_2 = 0 \).- \( (0)e^0 - e^0 + C_3 = 1 \), so \( C_3 = 2 \).

Step 5: Form the Function \( \mathbf{r}(t) \)

Substituting back, we have:- **i-component** is \( \frac{t^2}{2} + 1 \).- **j-component** is \( e^t \).- **k-component** is \( t e^t - e^t + 2 \). Combine them to obtain \( \mathbf{r}(t) = \left( \frac{t^2}{2} + 1 \right) \mathbf{i} + e^t \mathbf{j} + \left( t e^t - e^t + 2 \right) \mathbf{k} \).

Key Concepts

Integration by Parts

Integration by parts is a mathematical technique used to integrate products of functions. It is particularly useful when dealing with functions that are multiplied together, like in the k-component of our vector function. In general, the formula for integration by parts is given by:\[ \int u \, dv = uv - \int v \, du \]Let's break down the process as it applied to our exercise. We needed to integrate the k-component, which was given as \( t e^t \). This is a product of two terms: \( t \) and \( e^t \). To proceed, we selected:

• \( u = t \)
• \( dv = e^t \, dt \)

This means:

• \( du = dt \)
• \( v = e^t \)

Plugging these into the integration by parts formula gave us:

• \( uv = t e^t \)
• \( \int v \, du = \int e^t \, dt = e^t \)

Therefore, the integral of \( t e^t \) using integration by parts is:\[ t e^t - e^t \] This method not only helps in finding integrals of products but also simplifies otherwise difficult calculations.

Parametric Equations

Parametric equations allow us to express a set of related quantities as explicit functions of one or more independent variables, called parameters. In vector calculus, parametric equations are often used to describe curves in space.For the problem at hand, we have the velocity vector \( \mathbf{r}'(t) = t \mathbf{i} + e^t \mathbf{j} + t e^t \mathbf{k} \). This tells us how the position changes with respect to time. To find the position vector \( \mathbf{r}(t) \), we need to integrate each component separately with respect to the parameter \( t \).Each component is treated like an individual parametric equation. Integration of these separate equations gives us:

• For the **i-component**: \( \frac{t^2}{2} + C_1 \)
• For the **j-component**: \( e^t + C_2 \)
• For the **k-component**: \( t e^t - e^t + C_3 \)

These parametric equations provide a compact representation of the path traced by a point moving along the curve defined by \( \mathbf{r}(t) \). The parameter \( t \) here, represents the time variable that dictates how the system evolves.

Integration Constants

Integration constants arise whenever we perform indefinite integration. This is due to the fact that differentiation of a constant yields zero, so essential information can be lost in the integration process. In our problem, the integration resulted in a constant for each component after finding the indefinite integrals for the **i**, **j**, and **k** components. Denoted as \( C_1 \), \( C_2 \), and \( C_3 \), these constants need to be determined using additional information:We use the initial condition \( \mathbf{r}(0) = \mathbf{i} + \mathbf{j} + \mathbf{k} \) to find these constants. This step involves substituting \( t = 0 \) into the equations for each component:

• **i-component**: \( \frac{0^2}{2} + C_1 = 1 \), resulting in \( C_1 = 1 \).
• **j-component**: \( e^0 + C_2 = 1 \), leading to \( C_2 = 0 \).
• **k-component**: \( 0 \cdot e^0 - e^0 + C_3 = 1 \), providing \( C_3 = 2 \).

By determining these constants, we ensure that our solution satisfies the given initial conditions and accurately describes the system. These constants are an integral part of solving differential equations, as they allow us to pinpoint the precise curve amongst the infinitely many possible solutions.