Question

Solve the initial value problem y' =(x+y-1)² with y(0) = 0. To solve this, we should use the substitution u =____________ u' = ____________ Enter derivatives using prime notation (e.g., you would enter y' for dy/dx). After the substitution from the previous part, we obtain the following differential equation in x, u, u'____________ The solution to the original initial value problem is described by the following equation in x, y____________

Answer 1

To solve the initial value problem y' =(x+y-1)² with y(0) = 0, we can use the substitution u = (x+y-1) and solve the resulting differential equation. The solution to the original initial value problem is y = c, where c is a constant.

Step-by-step explanation:

To solve the initial value problem y' =(x+y-1)² with y(0) = 0, we can use the substitution u = (x+y-1). By taking the derivative of u with respect to x, we get u' = 1+y'. After the substitution, we obtain the differential equation ((u+1)²)u' = u². This equation can be solved by separating variables and integrating.

Next, we solve the equation using the substitution v= u+1. The equation becomes v²-2v+v' = v²-1. Simplifying further, we get v' = 1. Integrating this equation, we get v = x+c, where c is a constant of integration.

Finally, we substitute back for v and solve for u and y. Since v = u+1, the equation becomes u+1 = x+c, and u = x+c-1. Substituting u into the original substitution equation, we have x+y-1 = x+c-1, which gives y = c. Therefore, the solution to the initial value problem is y = c, where c is a constant.