Find the particular antiderivative that satisfies the following conditions: dy/dx=8x^-2+6x^-1-3 where y(1)=7
To find the particular antiderivative that satisfies the given conditions, we need to integrate the expression dy/dx = 8x^(-2) + 6x^(-1) - 3 with respect to x.
Step 1: Integrate each term of the expression.
The antiderivative of 8x^(-2) is 8 * (x^(-2 + 1)) / (-2 + 1) = -8x^(-1).
The antiderivative of 6x^(-1) is 6 * (x^(-1 + 1)) / (-1 + 1) = 6 ln(x).
The antiderivative of -3 is -3x.
Step 2: Combine the antiderivatives.
Now that we have the antiderivatives of each term, we can combine them to find the general antiderivative:
y(x) = -8x^(-1) + 6 ln(x) - 3x + C
Here, C represents the constant of integration.
Step 3: Use the given initial condition to determine the value of C.
We are given that y(1) = 7, which means when x = 1, y = 7. Substituting these values into the general antiderivative equation:
7 = -8(1)^(-1) + 6 ln(1) - 3(1) + C
Simplifying, we have:
7 = -8 + C
C = 7 + 8
C = 15
Step 4: Substitute the value of C back into the general antiderivative to find the particular antiderivative.
The particular antiderivative that satisfies the given conditions is:
y(x) = -8x^(-1) + 6 ln(x) - 3x + 15