(D^2 + 1) y = sec^2 (x) tan (x)
solve for Yp.
To solve for Yp in the equation (D^2 + 1)y = sec^2(x)tan(x), we can use the method of undetermined coefficients. This method involves assuming a particular form for the solution and substituting it back into the original equation to determine the coefficients.
1. Assume that Yp has the form:
Yp = Asec^2(x)tan(x)
Where A is the coefficient we want to determine.
2. Differentiate Yp twice with respect to x to find the derivatives:
Yp' = 2Asec^2(x)tan(x)sec^2(x) + Asec^2(x)sec^2(x)
= 3Asec^4(x)tan(x)
Yp'' = d/dx (3Asec^4(x)tan(x))
= 12Asec^4(x)tan^2(x)sec^2(x) + 3Asec^4(x)sec^2(x)
= 3Asec^4(x)(4tan^2(x) + 1)
3. Substitute the derivatives back into the original equation (D^2 + 1)y = sec^2(x)tan(x):
(D^2 + 1)Yp = sec^2(x)tan(x)
(D^2 + 1)(Asec^2(x)tan(x)) = sec^2(x)tan(x)
(D^2(Asec^2(x)tan(x)) + Asec^2(x)tan(x) = sec^2(x)tan(x)
3Asec^4(x)(4tan^2(x) + 1) + Asec^2(x)tan(x) = sec^2(x)tan(x)
Simplifying the equation:
12Asec^4(x)tan^2(x) + 3Asec^4(x) + Asec^2(x)tan(x) = sec^2(x)tan(x)
4. Equating the coefficients of like terms on both sides:
12Asec^4(x)tan^2(x) + 3Asec^4(x) + Asec^2(x)tan(x) = sec^2(x)tan(x)
Comparing the coefficients, we have:
12A = 0 => A = 0
3A = 1 => A = 1/3
A = 0 does not satisfy the equation, so A = 1/3.
Therefore, the particular solution Yp is:
Yp = (1/3)sec^2(x)tan(x)