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An initial-value problem (IVP) in differential equations is one where we have not only the differential equation but also an initial condition that specifies the value of the unknown function at a particular point. For example, in our problem, we need to solve the differential equation given by: \( y^{\text{prime}}=x \text{sin} x \) with the initial condition \( y(0)=3 \). The initial condition ensures a unique solution to the differential equation. This unique solution is found by first solving the differential equation generally and then applying the specific initial value to find the particular constant of integration.

Integration by parts is a useful technique for solving integrals where the standard methods (like substitution) don't easily apply. The formula for integration by parts is: \ \( \ {\int} u \, dv = uv - \int v \, du \). Let's apply this to our integral \( \int x \text{sin} x dx \). In our case, we set: \ \(u = x \)\t, then \( du = dx \)\t, \(dv = \text{sin} x \, dx \)\t, and \(v = -\text{cos} x \).\ Substituting these into our formula, we get: \ \( \int x \text{sin} x \, dx = -x \text{cos} x - \int -\text{cos} x \, dx \). This simplifies further to: \ \( -x \text{cos} x + \text{sin} x + C \), where \( C \) is the constant of integration. This is how we integrate a product of functions, which is often useful in solving differential equations.

Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives. These are distinct from partial differential equations, which involve multiple variables. ODEs can be either 'ordinary' or 'partial' – in this case, we are dealing with an ordinary differential equation, specifically a first-order ODE. The form of the equation we are solving here is \( y^{\text{prime}} = f(x) \), where \( f(x) = x \text{sin} x \). First-order indicates that the highest derivative is the first derivative (\( y^{\text{prime}} \)). Solving first-order ODEs often involves: identifying the type of ODE, performing an integration, and applying initial conditions as given.