Question

Find the particular antiderivative that satisfies the

following conditions: dy dx y= = The = 4 - 8x; y(0) = 5.
Find the particular antiderivative that satisfies the following conditions: \[ \frac{d y}{d x}=4-8 x ; \quad y(0)=5 \] \[ y= \]

Answer 1

To find the particular antiderivative of \(\frac{dy}{dx}\) that satisfies the given conditions \(\frac{dy}{dx} = 4 - 8x\) and \(y(0) = 5\), you'll need to integrate the right-hand side of the equation with respect to \(x\) and then apply the initial condition.

First, integrate \(\frac{dy}{dx} = 4 - 8x\) with respect to \(x\):

\[

\int (4 - 8x) \, dx = 4x - 4x^2 + C,

\]

where \(C\) is the constant of integration.

Now, you have the antiderivative of \(\frac{dy}{dx}\) as \(4x - 4x^2 + C\). To find the particular antiderivative that satisfies \(y(0) = 5\), plug in \(x = 0\) and set it equal to 5:

\[

4(0) - 4(0)^2 + C = 0 + 0 + C = C.

\]

So, \(C = 5\).

Therefore, the particular antiderivative that satisfies the given conditions is:

\[y = 4x - 4x^2 + 5.\]

Explanation: