Find the particular antiderivative that satisfies the following conditions:
dy/dx=7(x^−2)+5(x^−1)−2; y(1)=4.
y' = 7/x^2 + 5/x - 2
y = -7/x + 5lnx - 2x + C
so now you have
-7 + 0 - 2 + C = 4
C = 13
y = -7/x + 5lnx - 2x + 13
thank you!
To find the particular antiderivative that satisfies the given conditions, we can use the method of integrating factors. The integrating factor is given by e^(integral[P(x)dx]), where P(x) is the coefficient of the highest derivative term in the equation. In this case, P(x) = 7(x^-2) + 5(x^-1) - 2.
First, let's find the integral of P(x) with respect to x:
Integral[P(x)dx] = Integral[(7(x^-2) + 5(x^-1) - 2)dx]
To solve this integral, we can split it into three separate integrals:
Integral[7(x^-2)dx] + Integral[5(x^-1)dx] - Integral[2dx]
Integrating each term separately, we get:
(7 * integral[x^(-2)dx]) + (5 * integral[x^(-1)dx]) - (2 * integral[dx])
= [7 * (-x^(-1))] + [5 * ln|x|] - [2 * x] + C
= -7/x + 5ln|x| - 2x + C
Now that we have the integral of P(x), let's find the integrating factor:
e^(integral[P(x)dx]) = e^[-7/x + 5ln|x| - 2x + C]
= e^(-7/x) * e^(5ln|x|) * e^(-2x) * e^C
Next, let's simplify the integrating factor:
e^(5ln|x|) = |x|^5
e^C = C (since it is a constant)
Therefore, the integrating factor is:
I(x) = |x|^5 * e^(-7/x) * e^(-2x) * C
Now, let's use the integrating factor to solve the given differential equation:
Multiplying both sides of the differential equation dy/dx = 7(x^-2) + 5(x^-1) - 2 by the integrating factor I(x), we get:
|y|^5 * e^(-7/x) * e^(-2x) * C * dy/dx = |x|^5 * e^(-7/x) * e^(-2x) * C * [7(x^-2) + 5(x^-1) - 2]
Simplifying, we have:
|y|^5 * e^(-7/x) * e^(-2x) * dy = |x|^5 * e^(-7/x) * e^(-2x) * [7(x^-2) + 5(x^-1) - 2] * dx
Integrating both sides, we get:
Integral[|y|^5 * e^(-7/x) * e^(-2x) * dy] = Integral[|x|^5 * e^(-7/x) * e^(-2x) * [7(x^-2) + 5(x^-1) - 2] * dx]
Simplifying further, we have:
Integral[|y|^5 * e^(-7/x) * e^(-2x) * dy] = Integral[|x|^5 * e^(-7/x) * e^(-2x) * [7/x^2 + 5/x - 2] * dx]
Integrating each side separately, we get:
(1/6) * |y|^6 * e^(-7/x) * e^(-2x) = Integral[|x|^6 * e^(-7/x) * e^(-2x) * [7/x^2 + 5/x - 2] * dx]
To solve the definite integral, we need the limits of integration. In this case, y(1) = 4. Substituting the limits of integration, we get:
(1/6) * [4^6 * e^(-7/1) * e^(-2(1))] = Integral[|x|^6 * e^(-7/x) * e^(-2x) * [7/x^2 + 5/x - 2] * dx]
Simplifying the left side, we get:
(1/6) * [4096 * e^(-7) * e^(-2)] = Integral[|x|^6 * e^(-7/x) * e^(-2x) * [7/x^2 + 5/x - 2] * dx]
Now that we have solved the integral, we can find the particular antiderivative:
(1/6) * [4096 * e^(-7) * e^(-2)] = F(b) - F(a)
where F(x) is the particular antiderivative of the given function and a = 1, b is the unknown value.
Unfortunately, with the given conditions, we cannot determine the specific values for a and b. Therefore, we cannot find the particular antiderivative that satisfies the given conditions.