Question

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy/dx = f(Ax + By + C), which is given in (5) of Section 2.5.

dy/dx = (x + y + 1)²

Answer 1

To solve the given differential equation dy/dx = (x + y + 1)², we can use the substitution method. The solution is y = tan(x + C) - x - 1.

Step-by-step explanation:

To solve the given differential equation dy/dx = (x + y + 1)², we can use the substitution method. Let's substitute u = x + y + 1. Taking the derivative of u with respect to x, we have du/dx = 1 + dy/dx. Rearranging the equation, we get dy/dx = du/dx - 1. Substituting this into the original equation, we have du/dx - 1 = u². This is a separable differential equation, which can be solved by separating the variables and integrating.

Let's simplify the equation: du/(u² + 1) = dx. To integrate the left side, we can use the arctan function. Integrating both sides, we have arctan(u) = x + C, where C is the constant of integration. Solving for u, we get u = tan(x + C).

Now, substituting back u = x + y + 1, we have x + y + 1 = tan(x + C). Rearranging the equation, we get y = tan(x + C) - x - 1. This is the solution to the given differential equation.