Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 2x − 7, y(2) = 0

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Identify the given differential equation and initial condition: $\frac{dy}{dx} = 2x - 7$ with $y(2) = 0$.

Integrate the right-hand side of the differential equation with respect to $x$ to find the general solution for $y$. That is, compute $y = \int (2x - 7) \, dx$.

Perform the integration: integrate \$2x$ to get $x^2$ and integrate $-7$ to get $-7x$, then add the constant of integration $C$ to write $y = x^2 - 7x + C$.

Use the initial condition $y(2) = 0$ to find the value of the constant $C$ by substituting $x = 2$ and $y = 0$ into the general solution.

Solve the resulting equation for $C$ and write the particular solution $y = x^2 - 7x + C$ with the found value of $C$.

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주요 개념

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

Differential Equations

A differential equation relates a function to its derivatives. In this problem, dy/dx = 2x − 7 expresses the rate of change of y with respect to x. Solving it means finding a function y(x) whose derivative matches the given expression.

Initial Value Problem (IVP)

An initial value problem specifies both a differential equation and a condition that the solution must satisfy at a particular point, here y(2) = 0. This condition helps determine the unique solution among infinitely many possible antiderivatives.

Integration to Solve First-Order ODEs

To solve dy/dx = 2x − 7, integrate the right-hand side with respect to x to find y(x). After integration, use the initial condition to solve for the constant of integration, ensuring the solution fits the given initial value.

Initial Value ProblemsSolve the initial value problems in Exercises 71–90.dy/dx = 2x − 7, y(2) = 0