Examples of differential equation in the following topics:
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- Differential equations can be used to model a variety of physical systems.
- Differential equations are very important in the mathematical modeling of physical systems.
- Many fundamental laws of physics and chemistry can be formulated as differential equations.
- Conduction of heat is governed by another second-order partial differential equation, the heat equation .
- Give examples of systems that can be modeled with differential equations
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- Differential equations are solved by finding the function for which the equation holds true.
- Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
- Solving the differential equation means solving for the function $f(x)$.
- The "order" of a differential equation depends on the derivative of the highest order in the equation.
- You can see that the differential equation still holds true with this constant.
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- In the previous atom, we learned that a second-order linear differential equation has the form:
- When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
- In general, the solution of the differential equation can only be obtained numerically.
- Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
- Identify when a second-order linear differential equation can be solved analytically
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- A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
- Linear differential equations are of the form $Ly = f$, where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and the right hand side $f$ is a given function of the same nature as $y$ (called the source term).
- where $D$ is the differential operator $\frac{d}{dt}$ (i.e.
- When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
- A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.
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- Separable differential equations are equations wherein the variables can be separated.
- Non-linear differential equations come in many forms.
- A separable equation is a differential equation of the following form:
- The original equation is separable if this differential equation can be expressed as:
- This is the easiest variety of differential equation to solve.
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- Direction fields and Euler's method are ways of visualizing and approximating the solutions to differential equations.
- Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
- The slope field is traditionally defined for differential equations of the following form:
- Then, from the differential equation, the slope to the curve at $A_0$ can be computed, and thus, the tangent line.
- Describe application of direction fields and Euler's method to approximate the solutions to differential equations
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- A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
- The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:
- Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
- Linear differential equations are of the form:
- where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and $f$ is a given function of the same nature as y (called the source term).
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- A second-order linear differential equation can be commonly found in physics, economics, and engineering.
- Examples of homogeneous or nonhomogeneous second-order linear differential equation can be found in many different disciplines, such as physics, economics, and engineering.
- The equation of motion is given as:
- Therefore, we end up with a homogeneous second-order linear differential equation:
- Identify problems that require solution of nonhomogeneous and homogeneous second-order linear differential equations
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- Parametric equations are a set of equations in which the coordinates (e.g., $x$ and $y$) are expressed in terms of a single third parameter.
- This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise.
- Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations.
- If one of these equations can be solved for $t$, the expression obtained can be substituted into the other equation to obtain an equation involving $x$ and $y$ only.
- In some cases there is no single equation in closed form that is equivalent to the parametric equations.
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- The differential $dy$ is defined by:
- The notation is such that the equation
- The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or a particular analytical significance if the differential is regarded as a linear approximation to the increment of a function.
- Higher-order differentials of a function $y = f(x)$ of a single variable $x$ can be defined as follows:
- Use implicit differentiation to find the derivatives of functions that are not explicitly functions of $x$