Question

Solve the initial value problem 2yy' +2 =y²+2x with y(0) = 4. a. To solve this, we should use the substitution u = _______________ With this subsitution, y = ______________ y' = _______________ 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′ : __________________ c. 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 2yy' + 2 = y² + 2x with y(0) = 4, we can make the substitution u = y² + 2. After the substitution, the differential equation becomes (1/2)(u - 2)u' + 2 = u. The solution to the original initial value problem is described by the equation x = (y² + 2)x - 2√(y² + 2) + C.

Step-by-step explanation:

To solve the initial value problem 2yy' + 2 = y² + 2x with y(0) = 4, we can make the substitution u = y² + 2. With this substitution, y = sqrt(u - 2) and y' = (1/2)(u - 2)^(-1/2)u'.

After the substitution, the differential equation becomes (1/2)(u - 2)u' + 2 = u. Rearranging, we get u' = 2/(u - 2).

The solution to the original initial value problem is described by the equation x = (y² + 2)x - 2√(y² + 2) + C, where C is a constant determined by the initial condition y(0) = 4.