Question

Solve the IVP. y′′+y=cosx;y(0)=1,y′(0)=−1

Answer 1

To solve the given initial value problem (IVP) y′′+y=cosx, differentiate the given equation twice and substitute initial conditions. Solve the resulting differential equation, write the general solution, and substitute the initial conditions to get the particular solution: y(x) = -sin(x) + cos(x).

Step-by-step explanation:

To solve the given initial value problem (IVP) y′′+y=cosx, we need to find the equation for the second derivative of y and then solve the resulting differential equation.

1. First, differentiate the given equation twice to find y′′.
2. Substitute the initial conditions y(0)=1 and y′(0)=−1 into the differential equation and the equation for y′′.
3. Now solve the resulting differential equation by assuming y is a sum of particular and homogeneous solutions, and apply the initial conditions to find the constant values.
4. Finally, write the general solution of the differential equation, using trigonometric identities, and substitute the initial conditions to find the particular solution.

The solution to the given initial value problem is y(x) = -sin(x) + cos(x).