Differential Equations: Fundamentals

What defines the 'order' of a differential equation?

The order of a differential equation is determined by the highest-order derivative present in the equation.

What is the primary difference between an Ordinary Differential Equation (ODE) and a Partial Differential Equation (PDE)?

An ODE contains derivatives with respect to only one independent variable, whereas a PDE contains partial derivatives with respect to two or more independent variables.

What are the two main conditions for a differential equation to be considered linear?

1. The dependent variable and all its derivatives are of the first degree. 2. No products of the dependent variable or its derivatives exist; coefficients depend only on the independent variable.

What is the general solution to the first-order differential equation dy/dt = ky, representing exponential growth or decay?

y(t) = Ce^{kt}, where C is a constant determined by initial conditions.

For a second-order linear homogeneous ODE with constant coefficients, ay'' + by' + cy = 0, what is the associated characteristic equation?

ar^2 + br + c = 0, where r represents the roots used to find the general solution.

Define the Laplace transform of a function f(t) for t >= 0.

L{f(t)} = F(s) = integral from 0 to infinity of e^{-st} f(t) dt.

Under what condition is a first-order differential equation considered 'separable'?

An equation is separable if it can be written in the form dy/dx = g(x)h(y), allowing all terms involving y to be moved to one side and all terms involving x to the other.

What is the formula for the integrating factor mu(x) used to solve the linear first-order ODE y' + P(x)y = Q(x)?

mu(x) = exp(integral P(x) dx).

According to Picard's Existence and Uniqueness Theorem, what conditions on f(x, y) in y' = f(x, y) guarantee a unique solution near an initial point (x0, y0)?

Both the function f(x, y) and its partial derivative with respect to y, df/dy, must be continuous in a region containing the initial point.

State the differential equation for Newton's Law of Cooling, where T is object temperature, Tm is ambient temperature, and k is a constant.

dT/dt = -k(T - Tm).