To solve the given differential equation using the method of variation of parameters, we will follow these steps:

Step 1: Find the complementary solution:
First, we need to find the complementary solution to the homogeneous equation (D^2 - 1)y = 0. The characteristic equation is r^2 - 1 = 0, which gives us the roots r = 1 and r = -1. Therefore, the complementary solution is yc = C1e^x + C2e^(-x), where C1 and C2 are arbitrary constants.

Step 2: Find the particular solution:
To find the particular solution, we assume that it can be written as yp = u1(x)e^x + u2(x)e^(-x), where u1(x) and u2(x) are functions to be determined.

Step 3: Substitute the particular solution into the differential equation:
Substituting yp into the original differential equation (D^2 - 1)y = e^(-x)sin(e^(-x)) + cos(e^(-x)), we get:
(D^2 - 1)(u1(x)e^x + u2(x)e^(-x)) = e^(-x)sin(e^(-x)) + cos(e^(-x))

Step 4: Differentiate the particular solution:
Differentiate yp with respect to x to find yp' and differentiate again to find yp'':
yp' = u1'(x)e^x + u1(x)e^x - u2'(x)e^(-x) - u2(x)e^(-x)
yp'' = u1''(x)e^x + 2u1'(x)e^x - u2''(x)e^(-x) - 2u2'(x)e^(-x)

Step 5: Substitute the derivatives into the differential equation:
Substituting yp, yp', and yp'' into the differential equation, we get:
(u1''(x)e^x + 2u1'(x)e^x - u2''(x)e^(-x) - 2u2'(x)e^(-x)) - (u1(x)e^x + u2(x)e^(-x)) = e^(-x)sin(e^(-x)) + cos(e^(-x))

Step 6: Simplify the equation:
Simplifying the equation, we get:
u1''(x)e^x + 2u1'(x)e^x - u2''(x)e^(-x) - 2u2'(x)e^(-x) - u1(x)e^x - u2(x)e^(-x) = e^(-x)sin(e^(-x)) + cos(e^(-x))

Step 7: Solve for the functions u1(x) and u2(x):
By comparing the coefficients of the terms with the same exponential factors, we can set up a system of equations to solve for u1(x) and u2(x).

Step 8: Integrate to find the particular solution:
Once we have determined u1(x) and u2(x), we can integrate them to find the particular solution yp.

Step 9: Find the general solution:
The general solution to the original differential equation is given by y = yc + yp, where yc is the complementary solution and yp is the particular solution.

By following these steps, you can solve the given differential equation using the method of variation of parameters.

Question

To solve the given differential equation using the method of variation of parameters, we will follow these steps:

Step 1: Find the complementary solution:
First, we need to find the complementary solution to the homogeneous equation (D^2 - 1)y = 0. The characteristic equation is r^2 - 1 = 0, which gives us the roots r = 1 and r = -1. Therefore, the complementary solution is yc = C1e^x + C2e^(-x), where C1 and C2 are arbitrary constants.

Step 2: Find the particular solution:
To find the particular solution, we assume that it can be written as yp = u1(x)e^x + u2(x)e^(-x), where u1(x) and u2(x) are functions to be determined.

Step 3: Substitute the particular solution into the differential equation:
Substituting yp into the original differential equation (D^2 - 1)y = e^(-x)sin(e^(-x)) + cos(e^(-x)), we get:
(D^2 - 1)(u1(x)e^x + u2(x)e^(-x)) = e^(-x)sin(e^(-x)) + cos(e^(-x))

Step 4: Differentiate the particular solution:
Differentiate yp with respect to x to find yp' and differentiate again to find yp'':
yp' = u1'(x)e^x + u1(x)e^x - u2'(x)e^(-x) - u2(x)e^(-x)
yp'' = u1''(x)e^x + 2u1'(x)e^x - u2''(x)e^(-x) - 2u2'(x)e^(-x)

Step 5: Substitute the derivatives into the differential equation:
Substituting yp, yp', and yp'' into the differential equation, we get:
(u1''(x)e^x + 2u1'(x)e^x - u2''(x)e^(-x) - 2u2'(x)e^(-x)) - (u1(x)e^x + u2(x)e^(-x)) = e^(-x)sin(e^(-x)) + cos(e^(-x))

Step 6: Simplify the equation:
Simplifying the equation, we get:
u1''(x)e^x + 2u1'(x)e^x - u2''(x)e^(-x) - 2u2'(x)e^(-x) - u1(x)e^x - u2(x)e^(-x) = e^(-x)sin(e^(-x)) + cos(e^(-x))

Step 7: Solve for the functions u1(x) and u2(x):
By comparing the coefficients of the terms with the same exponential factors, we can set up a system of equations to solve for u1(x) and u2(x).

Step 8: Integrate to find the particular solution:
Once we have determined u1(x) and u2(x), we can integrate them to find the particular solution yp.

Step 9: Find the general solution:
The general solution to the original differential equation is given by y = yc + yp, where yc is the complementary solution and yp is the particular solution.

By following these steps, you can solve the given differential equation using the method of variation of parameters.

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