Notes in Week 6 - ODE's

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Published	11/26/2024	Ordinary Differential Equations (ODEs) describe various real-life phenomena by using {{c1::differential equations}} to represent rates of change.
Published	11/26/2024	The process of converting a physical problem into a mathematical model is known as {{c1::modeling}}.
Published	11/26/2024	The equation \( y' = \cos x \) is an example of an {{c1::ordinary differential equation (ODE)}}.
Published	11/26/2024	The term \( y' \) in ODEs denotes the derivative \( \frac{dy}{dx} \), representing the rate of change of \( y \) with respect to {{c1::x}}.
Published	11/26/2024	The constant of integration in an ODE solution accounts for initial conditions, often represented as {{c1::C}}.
Published	11/26/2024	Malthus's law describes exponential {{c1::growth}} and {{c2::decay}} using differential equations.
Published	11/26/2024	A {{c1::general solution}} of an ODE includes arbitrary constants.
Published	11/26/2024	An {{c1::initial value problem (IVP)}} combines a differential equation with initial conditions to determine a unique solution.
Published	11/26/2024	In the IVP \( y' = 3y, \ y(0) = 5.7 \), the particular solution is {{c1::\( y(x) = 5.7e^{3x} \)}}.
Published	11/26/2024	The numerical approach using Euler's Method involves incrementing by steps, defined as {{c1::\( y_{i+1} = y_i + slope \cdot h \)}}.
Published	11/26/2024	A direction field (or slope field) helps visualize solutions to ODEs by showing {{c1::the direction of solution curves}}.
Published	11/26/2024	The slope field is made by drawing short line segments, called {{c1::lineal elements}}, in the \(xy\)-plane.
Published	11/26/2024	The Runge-Kutta Method provides a way to solve ODEs without {{c1::calculating higher derivatives}}.
Published	11/26/2024	The fourth-order Runge-Kutta method (RK4) approximates solutions by using {{c1::a weighted sum of slopes}}.
Published	11/26/2024	A first-order ODE is said to be linear if it can be written in the form {{c1::\( y' + p(x)y = r(x) \)}}.
Published	11/26/2024	In a linear ODE, {{c1::\( r(x) \)}} is known as the {{c2::input}}, while {{c3::\( y(x) \)}} is called the {{c4::output or response}}.
Published	11/26/2024	The Bernoulli equation is a type of ODE that can be made linear when {{c1::a = 0}} or {{c2::a = 1}}.
Published	11/26/2024	The general form of the Bernoulli equation is {{c1::\( y' + p(x)y = g(x)y^a \)}}.
Published	11/26/2024	A homogeneous linear ODE has the form {{c1::\( y' + p(x)y = 0 \)}}.
Published	11/26/2024	Numerical approaches to solving ODEs are often essential because many ODEs lack {{c1::closed-form solutions}}.
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