Differential Equations Flashcard

Front

What is a differential equation?

Back

A differential equation is a mathematical equation that relates a function with its derivatives. It describes how a quantity changes in relation to another quantity.

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What does the notation dy/dx represent?

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The notation dy/dx represents the derivative of y with respect to x, indicating the rate of change of y as x changes.

Front

What is the general solution of a first-order linear differential equation?

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The general solution of a first-order linear differential equation is of the form y = Ce^{rac{1}{m}x} + f(x), where C is a constant and f(x) is a particular solution.

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What is the purpose of initial conditions in solving differential equations?

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Initial conditions are used to find the particular solution of a differential equation by providing specific values for the function and its derivatives at a given point.

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How do you solve a separable differential equation?

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To solve a separable differential equation, you separate the variables by rearranging the equation into the form g(y)dy = h(x)dx, then integrate both sides.

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What is the slope field of a differential equation?

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A slope field is a graphical representation of the solutions of a differential equation, showing the slope of the solution curves at various points in the plane.

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What is the significance of the constant 'C' in the general solution of a differential equation?

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The constant 'C' represents the family of solutions to the differential equation, indicating that there are infinitely many solutions differing by a constant.

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What is the method of integrating factors?

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The method of integrating factors is a technique used to solve first-order linear differential equations by multiplying the equation by a specific function to make it easier to integrate.

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What is a particular solution?

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A particular solution is a specific solution to a differential equation that satisfies given initial or boundary conditions.

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What is the relationship between a differential equation and its solution?

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The solution of a differential equation is a function that satisfies the equation, meaning it produces the correct derivatives when substituted back into the equation.