Examples

Indefinite vector integral calculation

Indefinite vector integral calculation

Problem: Compute the indefinite vector integral: $\int (e^t,t^{\frac{1}{2}},\cos (t))dt$.

Solution: The vector integral is given by computing ordinary integrals in each component. We have:

$$\begin{array}{rl}\int (e^t,t^{\frac{1}{2}},\cos (t))dt& =\left(\int e^{t}dt,\int t^{\frac{1}{2}}dt,\int \cos (t)dt\right)\\ & =\left(e^t+c_1,\frac{2}{3}{t}^{\frac{3}{2}}+c_2,\sin (t)+c_3\right)\\ & =\left(e^t,\frac{2}{3}{t}^{\frac{3}{2}},\sin (t)\right)+\mathbf{c}.\end{array}$$

Particle position from velocity - differential equation

Particle position from velocity: differential equation

Problem: Suppose that the path of a particle is given by $\mathbf{r}(t)$ and that its velocity satisfies ${\mathbf{r}}^{\prime }(t)=(1-6\sin (3t),4t,e^t)$, and suppose the particle passes through the point $(4,1,2)$ at $t=0$. Find $\mathbf{r}(t)$ and use your formula to find the position of the particle at $t=2$.

Solution: By taking the indefinite vector integral of both sides of ${\mathbf{r}}^{\prime }(t)=(1-6\sin (3t),4t,e^t)$, we obtain the general solution:

$$\mathbf{r}(t)=\left(t+2\cos (3t),2t^2,e^t\right)+\mathbf{c}.$$

We can solve for $\mathbf{c}$ by setting $\mathbf{r}(0)=(4,1,2)$:

$$(4,1,2)=(2,0,1)+\mathbf{c}$$

so $\mathbf{c}=(2,1,1)$ gives the particular solution satisfying the initial condition $\mathbf{r}(0)=(4,1,2)$.