y’=-6y+7

To solve the differential equation y' = -6y + 7, we can use the method of separation of variables.

First, recognize that this is a first-order linear differential equation. Rewrite it in the standard form of a linear differential equation as follows:

y' + 6y = 7

Now, separate the variables (y and t) by moving y terms to one side and t terms to the other side:

dy/dt + 6y = 7

Next, we solve for y by integrating both sides of the equation. The general form of the solution is given by:

y(t) = e^(-6t) * (C + ∫e^(6t) * 7 dt)

Solving the integral on the right side:

y(t) = e^(-6t) * (C + (1/6) * e^(6t) * 7) + D

where C and D are constants of integration.

Finally, simplify and determine the values of C and D using initial conditions or additional information provided.