In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
2. y' = y²
b. y = -1/(x+3)

검증된 단계별 안내

Start by identifying the given function: \(y = -\frac{1}{x+3}\).

Find the derivative of \(y\) with respect to \(x\). Use the chain rule to differentiate \(y = - (x+3)^{-1}\), which gives \(y' = -(-1)(x+3)^{-2} \cdot 1 = \frac{1}{(x+3)^2}\).

Express \(y^2\) by squaring the original function: \(y^2 = \left(-\frac{1}{x+3}\right)^2 = \frac{1}{(x+3)^2}\).

Compare the derivative \(y'\) and \(y^2\) to check if they are equal: both are \(\frac{1}{(x+3)^2}\).

Since \(y' = y^2\), conclude that the function \(y = -\frac{1}{x+3}\) satisfies the differential equation \(y' = y^2\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Differential Equations

A differential equation relates a function to its derivatives. Solving it means finding a function that satisfies this relationship. In this problem, verifying a solution involves checking if the given function and its derivative satisfy the equation y' = y².

Derivative of a Function

The derivative represents the rate of change of a function with respect to its variable. To verify the solution, you must compute the derivative y' of the given function y = -1/(x+3) using differentiation rules, such as the chain or quotient rule.

Substitution and Verification

After finding the derivative y', substitute both y and y' into the differential equation to check if the equality holds. This process confirms whether the given function is indeed a solution to the differential equation.

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.2. y' = y²b. y = -1/(x+3)