Final answer:
Solving the initial value problem involves a substitution technique, transforming the original differential equation into a new variable u, simplifying the equation, and integrating. After integrating and finding u(x), substitute back to find y as a function of x.
Step-by-step explanation:
To solve the initial value problem y′=(x+y−2)² with y(0)=0, we leverage a substitution technique. We can simplify the problem by introducing a new variable, u, defined as u = x + y - 2. This implies that u′ = y′ + 1 because du/dx = dy/dx + dx/dx. Incorporating this substitution into our original equation, we get u′ = u² + 1.
Next, we need to solve this new differential equation. We separate the variables and integrate each side to find u as a function of x. After finding u, we can then revert back to our original variables by substituting back x + y - 2 for u to find the relationship between x and y.
The solution to the original initial value problem is then represented by the equation obtained from integrating and back-substituting u to express y as a function of x.