Final answer:
To determine if the given equation is exact, we need to check if the partial derivatives of the terms with respect to y and t are equal. In this case, the equation is exact. To solve the equation, we can find the integrating factor and multiply the equation by it. Integrating both sides gives us the solution.
Step-by-step explanation:
To determine if the given equation is exact, we need to check if the partial derivatives of the terms with respect to y and t are equal. In this case, the partial derivative of e^t(7y - 2t) with respect to y is 7e^t and the partial derivative of (5 + 7e^t) with respect to t is 7e^t. Since these two derivatives are equal, the equation is exact.
To solve the equation, we can find the integrating factor by taking the partial derivative of the coefficient of dy with respect to y. In this case, the coefficient is 5 + 7e^t. Taking the partial derivative of this with respect to y, we get 0. Since the integrating factor is constant, we can choose any constant value for it. Let's choose 1 for simplicity.
Multiplying the original equation by the integrating factor, we get e^t(7y - 2t) dt + (5 + 7e^t) dy = 0. Integrating both sides gives us the solution: e^t(7y - 2t) + 5y + 7e^t = C, where C is the constant of integration.