Find the particular antiderivative that satisfies the following conditions

dy/dx=7x^[−2]+8x^[−1]−6;y(1)=2.

dy/dx = 7/x^2 + 8/x - 6

y = -7/x + 8 lnx - 6x + C
using the point (1,2) we get
-7 + 0 - 6 + C = 2
C = 15
so,
y = -7/x + 8 lnx - 6x + 15

Thank you

To find the particular antiderivative that satisfies these conditions, we need to integrate the given function and then solve for the constant of integration using the initial condition y(1) = 2.

Given: dy/dx = 7x^(-2) + 8x^(-1) - 6

To integrate the function, we'll treat each term separately:

∫(7x^(-2) + 8x^(-1) - 6)dx

= ∫7x^(-2) dx + ∫8x^(-1) dx - ∫6dx

Integrating each term gives:

= 7∫x^(-2) dx + 8∫x^(-1) dx - ∫6 dx

∫x^(-2) dx = -x^(-1)

∫x^(-1) dx = ln|x|

∫6 dx = 6x

Plugging these integrals back into the original equation:

= 7(-x^(-1)) + 8(ln|x|) - 6x + C

Simplifying further:

= -7/x + 8ln|x| - 6x + C

Now, we will use the initial condition y(1) = 2 to find the value of the constant C.

y(1) = -7/1 + 8ln|1| - 6(1) + C
2 = -7 + 8(0) - 6 + C
2 = -13 + C
C = 15

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

y = -7/x + 8ln|x| - 6x + 15

To find the particular antiderivative that satisfies the given conditions, we first need to integrate the given differential equation.

The given differential equation is: dy/dx = 7x^(-2) + 8x^(-1) - 6

To integrate dy/dx, we look for the antiderivative of each term separately.

The antiderivative of 7x^(-2) can be found using the power rule for integration:
∫ 7x^(-2) dx = 7 * ∫ x^(-2) dx = 7 * (-1/x) = -7/x

The antiderivative of 8x^(-1) can also be found using the power rule for integration:
∫ 8x^(-1) dx = 8 * ∫ x^(-1) dx = 8 * ln|x|

The antiderivative of -6 is simply -6x.

Now, let's combine all the antiderivatives:
∫ dy/dx dx = ∫ (7x^(-2) + 8x^(-1) - 6) dx = -7/x + 8ln|x| - 6x + C

Here, C is the constant of integration, which is determined by the initial condition y(1) = 2.

To find the particular antiderivative that satisfies the condition y(1) = 2, we substitute x = 1 and y = 2 into the general antiderivative expression:

-7/1 + 8ln|1| - 6(1) + C = -7 + 8(0) - 6 + C = -13 + C

Since y(1) = 2, we set -13 + C = 2 and solve for C:

C = 2 + 13
C = 15

Therefore, the particular antiderivative that satisfies the given conditions is:
-7/x + 8ln|x| - 6x + 15