The order of a differential equation is determined by the highest order derivative that appears in the equation.

A first-order differential equation involves the first derivative of the dependent variable with respect to the independent variable.

An initial value problem consists of a differential equation along with a set of initial conditions that specify the values of the dependent variable and its derivatives at a particular point.

Separable differential equations can be solved using the method of integrating factors, which involves multiplying both sides of the equation by a suitable integrating factor to make it exact.

The general solution of the given differential equation can be obtained by separating the variables and integrating both sides.

An exact differential equation is one that can be expressed as the total derivative of a function. In this case, the equation $y' + \frac{1}{y} = x$ can be written as $d(y + \ln|y|) = x dx$.

The integrating factor for the given differential equation can be found by multiplying both sides of the equation by a suitable function that makes the left-hand side exact.

Laplace transforms are often used to solve linear differential equations with constant coefficients because they convert the differential equation into an algebraic equation that is easier to solve.

The general solution of the given differential equation can be obtained by finding the roots of the characteristic equation $r^2 + 4 = 0$ and using them to construct the general solution.

Variation of parameters is a method used to solve non-homogeneous linear differential equations by introducing a set of new functions that depend on the independent variable.

The Laplace transform of the given function can be obtained using the formula $\mathcal{L}{t^n e^{at}} = \frac{n!}{(s-a)^{n+1}}$.

A system of differential equations consists of two or more differential equations that are solved simultaneously.

The general solution of the given system of differential equations can be obtained by finding the eigenvalues and eigenvectors of the coefficient matrix.

A boundary value problem consists of a differential equation along with a set of boundary conditions that specify the values of the dependent variable at specific points.

The method of characteristics is a technique used to find the characteristics of a partial differential equation, which are curves in the independent variable space along which the solution of the equation is constant.