Ordinary Differential Equations Overview

A complex number written in the form z = re^(iθ), where r is the magnitude and θ is the angle from the positive real axis (in radians).

A graphical representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

The point where a complex number intersects the unit circle on the Argand diagram.

A complex number in the form a + bi, where a and b are real numbers.

A complex number that satisfies the equation z^n = w, where n is a positive integer and w is a complex number.

The value found by adding all the real parts of the complex numbers in a set.

The value found by adding all the imaginary parts of the complex numbers in a set.

A complex number whose real part is zero.

A complex number that represents the amplitude and phase of a sinusoidal function. Its modulus gives the amplitude, and its argument gives the phase.

The phasor of the sinusoidal function y = A cos (ωt + φ) is Z = Aeiφ. The amplitude of the function is |Z| and the phase is arg(Z).

To find the phasor of a sinusoidal function given as a sine, you need to convert it to a cosine function using the identity sin(x) = cos(x - π/2).

Given a sinusoidal function y = A sin (ωt + φ), the phasor can be found directly as Z = Aei(φ - π/2).

Phasors allow us to represent sinusoidal functions as complex numbers. This representation is useful for manipulating and analyzing sinusoidal functions.

For any sinusoidal function with frequency ω, its phasor uniquely identifies it among all other functions with the same frequency.

Phasors provide a powerful tool for manipulating and analyzing sinusoidal functions.

The phasor representation of a sinusoidal function allows us to represent the function compactly and efficiently.

An equation that relates an unknown function and its derivatives to an independent variable. For example, y'' + y = e^x is an ODE.

The variable that the function depends on. In y(x), x is the independent variable.

The function that depends on the independent variable. In y(x), y is the dependent variable.

Newton's second law connects force (F), mass (m), and acceleration (which is the second derivative of position with respect to time) in an equation. This relationship can be expressed as a differential equation: F(x,t) = m(d^2x/dt^2).

Many real-life problems require finding solutions to ODEs. However, not all ODEs have exact solutions. We often need to resort to numerical methods to approximate the solutions.

Newton's second law of motion is often written as a differential equation because it relates the force acting on an object to its acceleration, which is the second derivative of position with respect to time. Force might depend on position and time.

Solving ODEs involves finding a function that satisfies the given equation. This function might describe motion, temperature distribution, or other physical phenomena.

Many ODEs don't have easy solutions and require numerical methods to find an approximate solution. These methods use computer algorithms to compute solutions step-by-step.

A second-order ordinary differential equation (ODE) is linear if it can be expressed in a specific form where the dependent variable (y) and its derivatives appear only multiplied by functions of the independent variable (x).

An arbitrary first-order linear ODE can be derived from the general form of a second-order linear ODE by setting the coefficient of the second derivative (a2(x)) to zero.

A function of x that multiplies the dependent variable (y) or its derivatives in a linear second-order ODE. These functions can be constant or variable.

A function of x that appears on the right-hand side of a linear second-order ODE. It represents an external force or influence on the system.

An ODE where the dependent variable (y) or its derivatives appear in the equation in a non-linear way. They are not simply multiplied by functions of x.

Non-linear ODEs are generally much more complex and challenging to solve compared to linear ODEs.

An example of a linear ODE is: dy/dt + y = sin(t). In this equation, the dependent variable y and its derivative only appear multiplied by functions of the independent variable t.

An example of a non-linear ODE is: dy/dx + e^y = sin(x). In this case, the dependent variable y appears within an exponential function (e^y).

A differential equation is called exact if it can be written in the form P(x,y)dx + Q(x,y)dy = 0 where the partial derivative of P with respect to y is equal to the partial derivative of Q with respect to x.

To identify an exact differential equation, check if the partial derivative of P(x,y) with respect to y equals the partial derivative of Q(x,y) with respect to x.

The general solution of an exact differential equation can be found by integrating P(x,y) with respect to x (treating y as a constant) and then adding a function of y, denoted as g(y). We then differentiate this result with respect to y and equate it to Q(x,y) to determine g(y). Finally, we combine the results to obtain the general solution.

An initial condition specifies the value of the dependent variable (usually y) at a specific value of the independent variable (usually x). It helps to pinpoint a particular solution within the family of solutions given by the general solution.

Given an initial condition, we can substitute the values of x and y into the general solution and solve for the integration constant C. This results in a unique solution that satisfies both the differential equation and the initial condition.

A first-order differential equation is an equation involving the first derivative of the unknown function. These equations often describe phenomena that change over time, and their solutions provide information about the behavior of the system.

A differential equation that describes a system where the rate of change is proportional to the current value is called a first-order linear homogeneous differential equation. The general solution is given by an exponential function.

The general solution of a first-order linear homogeneous differential equation is given by y(x) = Ce^(kx), where C is an arbitrary constant and k is a constant related to the proportionality factor.

An ordinary differential equation (ODE) is exact if it can be written in the form P(x, y) + Q(x, y) * dy/dx = 0, and the partial derivatives of P with respect to y and Q with respect to x are equal (∂P/∂y = ∂Q/∂x).

A function Ψ(x, y) that generates an exact ODE through its partial derivatives. ∂Ψ/∂x = P(x, y) and ∂Ψ/∂y = Q(x, y).

The condition for an ODE to be exact: ∂P/∂y = ∂Q/∂x, where P and Q are functions of x and y.

A line integral, where the integrand is the product of a function of x, y and dx, plus another function of x, y and dy. ∫(P(x, y) dx + Q(x, y) dy) over the curve C.

A theorem stating that the line integral of a vector field around a closed curve C is equal to the double integral of the curl of the vector field over the enclosed region D. ∫(P(x, y) dx + Q(x, y) dy) = ∫∫(∂Q/∂x - ∂P/∂y) dA

A mathematical method for solving an exact ODE, where the potential function is found by integrating P(x, y) with respect to x and Q(x, y) with respect to y, and then comparing the results to find a common potential function.

The method of rewriting an ODE in the form of an exact ODE by multiplying it by an appropriate integrating factor. The integrating factor makes the ODE satisfy the exactness condition.