Question

Let y be the solution of the initial value problem y" +y= - sin(2x), y(0) = 0, y' (0) = 0. The maximum value of y is

Answer 1

The maximum value of y for the given initial value problem is 0.

Step-by-step explanation:

In order to find the maximum value of y for the given initial value problem, we need to solve the differential equation y'' + y = -sin(2x) with the initial conditions y(0) = 0 and y'(0) = 0. To do this, we can solve the homogeneous equation y'' + y = 0 first, which has the general solution y = c1*cos(x) + c2*sin(x). Plugging this solution into the nonhomogeneous equation, we can find a particular solution for y. The general solution for y is the sum of the general solution of the homogeneous equation and the particular solution for y.

Next, we need to find the values of c1 and c2 using the initial conditions y(0) = 0 and y'(0) = 0. Substituting x = 0 and y = 0 into the equation y = c1*cos(x) + c2*sin(x), we find that c1 = 0. To find c2, we differentiate y = c1*cos(x) + c2*sin(x) with respect to x and substitute x = 0 and y' = 0. This gives us c2 = 0. Therefore, the particular solution for y is 0.

Now we can find the maximum value of y by finding the maximum value of the sum of the general solution of the homogeneous equation and the particular solution for y. Since both the general solution of the homogeneous equation and the particular solution for y are equal to 0, the maximum value of y is also 0.

Answer 2

To find the maximum value of the function y, which is the solution to the differential equation y" + y = -sin(2x), we need to determine the general and particular solution, apply initial conditions, and evaluate the function where its derivative is zero.

Step-by-step explanation:

To solve the given initial value problem, y" + y = -sin(2x), with initial conditions y(0) = 0 and y'(0) = 0, we need to find a particular solution to the non-homogeneous differential equation and combine it with the general solution to the corresponding homogeneous equation y" + y = 0. A particular solution to the given equation can be found using the method of undetermined coefficients or another appropriate technique. Once we have the general and particular solutions, we apply the initial conditions to find the specific solution that fits the problem.

The maximum value of y, which we can denote as ymax, will occur at a point where the derivative y' is zero because this represents a turning point in the function. By finding where y' is zero, we can evaluate the original function y at that point to determine ymax.