Let y(x) is the particular integer of the differential equation (D-1)*2*y=x*e^x*sin*x then?

Question

Answer

Particular solution of the differential equation

Solution:

Step 1: First, find the complementary function by solving the homogeneous equation (D-1)*2*y = 0. The solution to this equation is y_c = A*e^x, where A is a constant.
Step 2: To find the particular solution, assume y_p = B*x*e^x*sin(x), where B is a constant to be determined.
Step 3: Substitute y_p into the original differential equation to find the value of B. After simplification, you will get B = -1/2.
Step 4: Therefore, the particular solution is y_p = -1/2*x*e^x*sin(x).
Step 5: The general solution is the sum of the complementary function and the particular solution, y = y_c + y_p.
Step 6: Substituting the values of y_c and y_p, the general solution is y = A*e^x - 1/2*x*e^x*sin(x).

Conclusion:

Finally, the particular solution of the given differential equation is y(x) = A*e^x - 1/2*x*e^x*sin(x), where A is a constant determined by initial conditions if provided.