Question

Solve the initial value problem y′=(x+y−2)² with y(0)=0. a. To solve this, we should use the substitution u= [ ]help (formulas u′=[ ]​ help (formulas) ​ Enter derivatives using prime notation (e.g., you would enter y′ for dy/dx​ ) . b. After the substitution from the previous part, we obtain the following differential equation in x,u,u′. help (equations) c. The solution to the original initial value problem is described by the following equation in x,y. help (equations).

Answer 1

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.