Calc AB, Ch 9: Differential Equations

Front

What is a differential equation?

Back

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

Front

What does it mean to solve a differential equation?

Back

To solve a differential equation means to find a function (or a set of functions) that satisfies the equation.

Front

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

Back

The general form is dy/dx = f(x, y), where f is a function of x and y.

Front

What is the initial condition in a differential equation?

Back

An initial condition is a value that specifies the state of the system at a particular point, often given as y(x0) = y0.

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What is separation of variables?

Back

Separation of variables is a method for solving differential equations by rearranging the equation to isolate the variables on different sides.

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How do you apply separation of variables?

Back

To apply separation of variables, rewrite the equation in the form g(y) dy = h(x) dx, then integrate both sides.

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What is an integrating factor?

Back

An integrating factor is a function used to multiply a non-exact differential equation to make it exact, facilitating its solution.

Front

What is the solution to the differential equation dy/dx = 4x/y?

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The solution is y^2 = 2x^2 + C, where C is a constant determined by initial conditions.

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What does it mean for a solution to pass through a point?

Back

It means that the function satisfies the equation at that specific point, often used to find the constant in the solution.

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What is the range of a solution curve?

Back

The range of a solution curve is the set of possible output values (y-values) of the function for a given domain (x-values).