Solving initial value problems Find the solution of the following initial value problems.
g'(x) = 7x(x⁶ - 1/7); g(1) = 2

검증된 단계별 안내

Step 1: Recognize that this is an initial value problem, which involves finding a function g(x) such that its derivative matches the given expression g'(x) = 7x(x⁶ - 1/7) and satisfies the initial condition g(1) = 2.

Step 2: To find g(x), integrate the derivative g'(x). Set up the integral: ∫7x(x⁶ - 1/7) dx. Break this into two terms: ∫7x * x⁶ dx - ∫7x * (1/7) dx.

Step 3: Compute each integral separately. For the first term, ∫7x * x⁶ dx, rewrite it as ∫7x⁷ dx and use the power rule for integration: ∫xⁿ dx = (xⁿ⁺¹)/(n+1). For the second term, ∫7x * (1/7) dx, simplify to ∫x dx and apply the power rule.

Step 4: Combine the results of the two integrals to form the general solution g(x). Add the constant of integration C to account for the indefinite integral.

Step 5: Use the initial condition g(1) = 2 to solve for the constant C. Substitute x = 1 into the general solution g(x) and set it equal to 2. Solve for C to find the specific solution that satisfies the initial value problem.

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.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of the given problem, g'(x) = 7x(x⁶ - 1/7) indicates that we need to understand how to differentiate functions to find g(x) from its derivative.

Integration

Integration is the reverse process of differentiation and is used to find the original function from its derivative. To solve the initial value problem, we will integrate g'(x) to find g(x), which involves applying techniques such as substitution or integration by parts, depending on the complexity of the function.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation that includes a condition specifying the value of the function at a particular point. In this case, g(1) = 2 provides a specific value that allows us to determine the constant of integration after finding the general solution of g(x), ensuring that our solution satisfies the given condition.

Solving initial value problems Find the solution of the following initial value problems.g'(x) = 7x(x⁶ - 1/7); g(1) = 2