Question

Solve the given initial-value problem. (2y+2t−9)dt+(8y+2t−1)dy=0,y(−1)=2 Determine whether the given differential equation is exact. If it is exact, solve it.

Answer 1

The given differential equation is not exact. To solve it, we can use an integrating factor.

Step-by-step explanation:

This is a first-order linear ordinary differential equation. To determine if it is exact, we need to check if the partial derivatives satisfy the equality equation (M​y-Nx)dx+(N​x-My)dy = 0. In this case, M = 2y+2t−9, N = 8y+2t−1. Calculating the partial derivatives, we get My = 2 and Nx = 8, which are not equal. Therefore, the given differential equation is not exact.

To solve this type of problem, we can use an integrating factor. The integrating factor is defined as the exponential of the integral of (N​-M​y)/M. In this case, (N-M​y)/M = (8-2y)/(2y+2t−9). Integrating this expression with respect to y gives us the integrating factor.

Once we have the integrating factor, we multiply both sides of the differential equation by it. This transforms the equation into an exact one. We can then proceed to solve it using standard methods such as finding the partial derivatives, equating them to zero, and integrating.