(D^2 + 1) y = 1/(1 + sin x)
solve for Yp.
To solve for Yp, we need to find the particular solution to the given differential equation (D^2 + 1)y = 1/(1 + sin(x)). Here's how we can proceed:
Step 1: Rewrite the differential equation using the operator notation.
(D^2 + 1)y = 1/(1 + sin(x))
Step 2: Solve the homogeneous equation.
(D^2 + 1)y = 0
The characteristic equation for this homogeneous differential equation is m^2 + 1 = 0. Solving this equation, we find that the roots are m = ±i, where i is the imaginary unit.
Therefore, the homogeneous solution is y_h = c1*cos(x) + c2*sin(x), where c1 and c2 are arbitrary constants.
Step 3: Find the particular solution using the method of undetermined coefficients.
We can assume that the particular solution has the form Yp = A/(1 + sin(x)), where A is a constant to be determined.
Differentiating Yp with respect to x:
Yp' = -A*cos(x)/(1 + sin(x))^2
Differentiating Yp' with respect to x:
Yp'' = [2A*sin(x)*cos(x)]/(1 + sin(x))^3
Now substitute Yp, Yp', and Yp'' back into the differential equation:
(D^2 + 1)Yp = [2A*sin(x)*cos(x)]/(1 + sin(x))^3 + A/(1 + sin(x))
Simplify the equation:
[2A*sin(x)*cos(x)]/(1 + sin(x))^3 + A/(1 + sin(x)) = 1/(1 + sin(x))
Step 4: Equating the coefficients of like terms.
First, let's simplify the equation by multiplying both sides by (1 + sin(x))^3 to eliminate the denominators:
2A*sin(x)*cos(x) + A(1 + sin(x))^2 = 1
Now, equate the coefficients:
-2A*sin(x) = 0
A(1 + sin(x))^2 = 1
From the first equation, we can see that A = 0. However, substituting A = 0 into the second equation gives 0 = 1, which is not true.
Therefore, there is no particular solution Yp to the given differential equation.
Hence, the solution to the equation (D^2 + 1)y = 1/(1 + sin(x)) is the homogeneous solution y_h = c1*cos(x) + c2*sin(x), where c1 and c2 are arbitrary constants.