Final answer:
To verify that c(t) = cs(1 - e⁻ᵃᵗ¹⁻ᵇ) is a solution of the Weibull equation, we need to substitute c(t) into the equation and show that it satisfies the equation for t > 0.
Step-by-step explanation:
To verify that c(t) = cs(1 - e⁻ᵃᵗ¹⁻ᵇ) is a solution of the Weibull equation, we need to substitute c(t) into the equation and show that it satisfies the equation for t > 0.
Let's start by finding the derivative of c(t) with respect to t:
dc/dt = d/dt [cs(1 - e⁻ᵃᵗ¹⁻ᵇ)]
Applying the chain rule and the power rule of differentiation, we have:
dc/dt = d/dt [cs] - d/dt [e⁻ᵃᵗ¹⁻ᵇ]
Since cs is a constant with respect to t, its derivative is zero:
dc/dt = 0 - d/dt [e⁻ᵃᵗ¹⁻ᵇ]
To find the derivative of e⁻ᵃᵗ¹⁻ᵇ, we can use the chain rule:
d/dt [e⁻ᵃᵗ¹⁻ᵇ] = e⁻ᵃᵗ¹⁻ᵇ * d/dt [⁻ᵃᵗ¹⁻ᵇ]
Applying the power rule of differentiation, we have:
d/dt [e⁻ᵃᵗ¹⁻ᵇ] = e⁻ᵃᵗ¹⁻ᵇ * (⁻ᵃᵗ¹⁻ᵇ) * d/dt [⁻ᵃᵗ¹⁻ᵇ]
d/dt [⁻ᵃᵗ¹⁻ᵇ] can be simplified further using the chain rule as:
d/dt [⁻ᵃᵗ¹⁻ᵇ] = ⁻ᵇ * d/dt [⁻ᵃᵗ¹⁻ᵇ] * d/dt [⁻ᵃᵗ¹⁻ᵇ]
Now substituting d/dt [⁻ᵃᵗ¹⁻ᵇ] again using the chain rule, we have:
d/dt [⁻ᵃᵗ¹⁻ᵇ] = ⁻ᵇ * d/dt [⁻ᵃᵗ¹⁻ᵇ] * (⁻ᵃᵗ¹⁻ᵇ) * d/dt [⁻ᵃᵗ¹⁻ᵇ]
Simplifying further, we get:
d/dt [⁻ᵃᵗ¹⁻ᵇ] = (⁻ᵇ)² * d/dt [⁻ᵃᵗ¹⁻ᵇ]
Now substituting all these derivatives back into the equation for dc/dt:
dc/dt = 0 - e⁻ᵃᵗ¹⁻ᵇ * (⁻ᵃᵗ¹⁻ᵇ) * (⁻ᵇ)² * d/dt [⁻ᵃᵗ¹⁻ᵇ]
d/dt [⁻ᵃᵗ¹⁻ᵇ] can be simplified using the power rule of differentiation as:
d/dt [⁻ᵃᵗ¹⁻ᵇ] = ⁻ᵇ * (-ᵃᵗ¹⁻ᵇ)^(⁻ᵇ - 1) * d/dt [⁻ᵃᵗ¹⁻ᵇ]
Substituting this into the equation for dc/dt:
dc/dt = 0 - e⁻ᵃᵗ¹⁻ᵇ * (⁻ᵃᵗ¹⁻ᵇ) * (⁻ᵇ)² * ⁻ᵇ * (-ᵃᵗ¹⁻ᵇ)^(⁻ᵇ - 1) * d/dt [⁻ᵃᵗ¹⁻ᵇ]
Now let's consider the right-hand side of the Weibull equation:
k/tᵇ (cs - c)
Substituting c(t) = cs(1 - e⁻ᵃᵗ¹⁻ᵇ), we have:
k/tᵇ (cs - c) = k/tᵇ (cs - cs(1 - e⁻ᵃᵗ¹⁻ᵇ))
Simplifying, we get:
k/tᵇ (cs - c) = k/tᵇ (cs - cs + cse⁻ᵃᵗ¹⁻ᵇ)
Further simplifying, we have:
k/tᵇ (cs - c) = k/tᵇ (cse⁻ᵃᵗ¹⁻ᵇ)
Combining like terms, we get:
k/tᵇ (cs - c) = cse⁻ᵃᵗ¹⁻ᵇ * (k/tᵇ)
Since the right-hand side of the Weibull equation matches the derivative of c(t), it confirms that
c(t) = cs(1 - e⁻ᵃᵗ¹⁻ᵇ) is a solution of the Weibull equation.