(D^2 + 1) y = sec^2 2x
solve for Yp.
To solve for Yp, we need to find the particular solution (Yp) that satisfies the given differential equation.
The given differential equation is: (D^2 + 1)y = sec^2(2x), where D represents the differentiation operator.
To find Yp, we will use the method of undetermined coefficients. This method assumes that the particular solution can be expressed as a sum of terms, each corresponding to a known function.
First, let's determine the homogeneous solution (Yh) of the differential equation. To do this, we solve the associated homogeneous equation obtained by setting the right-hand side of the given equation equal to zero:
(D^2 + 1)y = 0
The characteristic equation corresponding to this homogeneous equation is: r^2 + 1 = 0.
Solving this quadratic equation, we find two imaginary roots: r1 = i and r2 = -i.
Therefore, the homogeneous solution Yh is given by: yh = c1 * cos(x) + c2 * sin(x), where c1 and c2 are arbitrary constants.
Next, we assume the particular solution Yp has the form:
Yp = A * sec^2(2x)
Here, A is a constant that we need to determine.
To find the particular solution Yp, we proceed as follows:
1. Take the first derivative of Yp:
Yp' = d/dx (A * sec^2(2x))
Using the chain rule, we have: Yp' = 2A * sec(2x) * tan(2x)
2. Take the second derivative of Yp:
Yp'' = d^2/dx^2 (A * sec^2(2x))
Using the chain rule and the derivative of sec(2x), we have: Yp'' = 2A * sec(2x) * tan(2x) * sec(2x) * tan(2x) + 2A * sec^2(2x) * sec^2(2x)
Simplifying this expression, we get: Yp'' = 2A * sec^2(2x) * (tan^2(2x) + sec^2(2x))
3. Substitute Yp, Yp', and Yp'' into the original differential equation:
(D^2 + 1)y = sec^2(2x)
(D^2 + 1)(A * sec^2(2x)) = sec^2(2x)
Now, let's substitute the expressions for Yp'' and Yp into the differential equation:
(2A * sec^2(2x) * (tan^2(2x) + sec^2(2x))) + A * sec^2(2x) = sec^2(2x)
Simplifying this equation, we can cancel out the common factor of sec^2(2x):
2A * (tan^2(2x) + sec^2(2x)) + A = 1
Now, we need to solve this equation for A:
2A * (tan^2(2x) + sec^2(2x)) + A = 1
Expanding the product and combining like terms, we get:
2A * tan^2(2x) + 2A * sec^2(2x) + A = 1
Using the trigonometric identity: tan^2(x) + 1 = sec^2(x), we can rewrite the equation as:
2A * sec^2(2x) + A * (sec^2(2x) + 1) = 1
3A * sec^2(2x) + A = 1
Factoring out A, we have:
A * (3 * sec^2(2x) + 1) = 1
Dividing both sides by 3 * sec^2(2x) + 1, we find:
A = 1 / (3 * sec^2(2x) + 1)
Now that we have determined the value of A, we can write the particular solution:
Yp = A * sec^2(2x)
Substituting the value of A, we get:
Yp = 1 / (3 * sec^2(2x) + 1) * sec^2(2x)
Simplifying further, we have:
Yp = sec^2(2x) / (3 * sec^2(2x) + 1)
Therefore, the particular solution Yp is equal to sec^2(2x) / (3 * sec^2(2x) + 1).