Solve the following initial value problem:𝑦=∫(3𝑥2−6)d𝑥,𝑦(1)=2
Question
Solve the following initial value problem:𝑦=∫(3𝑥2−6)d𝑥,𝑦(1)=2
Solution
Sure, let's solve the initial value problem step by step.
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First, we need to find the antiderivative of the function inside the integral. The antiderivative of 3x^2 - 6 is x^3 - 6x + C, where C is the constant of integration.
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So, y = ∫(3x^2 - 6)dx = x^3 - 6x + C.
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Now, we use the initial condition y(1) = 2 to find the value of C. Substituting x = 1 into the equation gives us 2 = 1^3 - 6*1 + C.
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Solving for C, we get C = 2 + 6 - 1 = 7.
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Therefore, the solution to the initial value problem is y = x^3 - 6x + 7.
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