Initial Value Problems

Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

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Identify the given differential equation from the problem: the slope of the curve at each point is given by \(\frac{dy}{dx} = 3\sqrt{x}\).

Rewrite the slope expression in terms of a power of \(x\) to make integration easier: \(\frac{dy}{dx} = 3x^{\frac{1}{2}}\).

Integrate both sides with respect to \(x\) to find the general form of \(y\): \(y = \int 3x^{\frac{1}{2}} \, dx\).

Perform the integration using the power rule for integrals: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.

Use the initial condition \(y(9) = 4\) to solve for the constant \(C\) by substituting \(x=9\) and \(y=4\) into the integrated function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Initial Value Problems

An initial value problem involves finding a function that satisfies a given differential equation and passes through a specific point, called the initial condition. This ensures a unique solution by fixing the constant of integration after solving the differential equation.

Differential Equations and Slope Fields

A differential equation relates a function to its derivatives. Here, the slope of the curve y = f(x) is given as a function of x, meaning dy/dx = 3√x. Understanding how to interpret and solve such equations is key to finding the original function.

Integration to Find the Original Function

To find y = f(x) from its derivative dy/dx, integrate the given slope function with respect to x. After integration, use the initial condition to solve for the constant of integration, yielding the specific curve passing through the given point.

Initial Value ProblemsFind the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.