Find both the general solution of the differential equation...

Find both the general solution of the differential equation and the solution with the given initial condition.
$$
\mathbf{r}^{\prime \prime}(t)=16 \mathbf{k}, \quad \mathbf{r}(0)=\langle 1,0,0\rangle, \quad \mathbf{r}^{\prime}(0)=\langle 0,1,0\rangle
$$

Key Concepts

Vector Differential Equations

This concept involves working with differential equations where the unknown function is a vector-valued function, often expressed as a combination of scalar functions for each coordinate. Such equations allow us to handle problems in higher dimensions and deal with phenomena like motion in space by breaking them down into component-wise equations.

Constant Acceleration

Constant acceleration refers to a scenario where the acceleration vector remains unchanged over time. In the context of differential equations, this means that the second derivative of the position vector is constant, which simplifies the process of integrating to find velocity and position, as each integration introduces a constant vector that can be determined using initial conditions.

Integration of Vector Functions

Integrating vector functions means performing the integration operation on each component of the vector separately. In solving second order vector differential equations, one integrates first to obtain the velocity vector and a second time to obtain the position vector, taking care to include the integration constants that represent the initial conditions of the system.

Initial Value Problems

An initial value problem (IVP) involves finding a unique solution to a differential equation that also satisfies given initial conditions. These conditions provide the necessary information to determine the constants of integration, ensuring that the solution matches the specified starting position and velocity, and thereby tailoring the general solution to the particular scenario described by the problem.

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