Question

In Exercises \(53-56,\) find \(\mathbf{r}(t)\) for the given conditions. $$ \mathbf{r}^{\prime}(t)=4 e^{2 t} \mathbf{i}+3 e^{t} \mathbf{j}, \quad \mathbf{r}(0)=2 \mathbf{i} $$

Short answer

The vector function \(\mathbf{r}(t)\) is \(2 e^{2 t} \mathbf{i} + 3 e^{t} \mathbf{j}\).

Step-by-step solution

Find the indefinite integral of \( \mathbf{r}'(t) \)

The integral is a process which reverses the operation of differentiation. So, we'll first calculate the integral of \(\mathbf{r}'(t)\). It will be \(\int 4 e^{2 t} \mathbf{i} dt + \int 3 e^{t} \mathbf{j} dt \).

Solve the integrals

After solving the integral, we get \(2 e^{2 t} \mathbf{i} + 3 e^{t} \mathbf{j} + C \), where C is the constant of integration, which is a vector in this case.

Apply the initial condition

From the initial condition, we know at t = 0, \(\mathbf{r}(t) = 2\mathbf{i}\). So, substituting this into \(\mathbf{r}(t)\), we find \(C = 2\mathbf{i}- 2\mathbf{i} = 0\).

Write out the final solution

Substitute the value of C from Step 3 into the equation from Step 2 to get the final solution for \(\mathbf{r}(t)\) which is \(2 e^{2 t} \mathbf{i} + 3 e^{t} \mathbf{j}\) .