Question

Consider the following differential equation to be solved by variation of parameters. y" + y = sec(θ) tan(θ) Find the complementary function of the differential equation. yc (θ) = Find the general solution of the differential equation. y(θ) = Solve the differential equation by variation of parameters. y" + y = sin^2(x) y(x) =

Answer 1

Complementary function of the differential equation: yc(θ) = A cos(θ) + B sin(θ)

General solution of the differential equation: y(θ) = yc(θ) + y particular(θ)

Solving the differential equation by variation of parameters: y(x) = C₁ cos(x) + C₂ sin(x) - (1/2) sin(2x)

Step-by-step explanation:

For the differential equation y" + y = sec(θ) tan(θ), the complementary function (yc) is derived from the homogeneous equation y" + y = 0, resulting in yc(θ) = A cos(θ) + B sin(θ). This represents the solution to the associated homogeneous differential equation. The general solution of the differential equation combines the complementary function with a particular solution (yp) obtained from the non-homogeneous part of the equation. The general solution is given by y(θ) = yc(θ) + y particular(θ).

Now, for solving the differential equation y" + y = sin^2(x), the complementary function yc(θ) = A cos(θ) + B sin(θ) remains the same. However, the particular solution for this equation, found using variation of parameters, is y particular(θ) = -(1/2) sin(2θ). Hence, the general solution for this equation becomes y(θ) = yc(θ) + y particular(θ) = C₁ cos(θ) + C₂ sin(θ) - (1/2) sin(2θ).

In summary, the complementary function represents the solution to the associated homogeneous equation, while the general solution incorporates both the complementary function and the particular solution obtained through methods like variation of parameters to solve non-homogeneous differential equations."".

Answer 2

The complementary function of the differential equation y'' + y = sec(θ) tan(θ) is given by yc(θ) = Ae^(kθ) + Be^(-kθ). The general solution of the differential equation can be found by combining the complementary function with the particular solution obtained using variation of parameters. To solve the differential equation y'' + y = sin^2(x) by variation of parameters, we find the complementary function, the particular solution, and then combine them to obtain the general solution.

Step-by-step explanation:

Complementary Function:

The complementary function of the differential equation y'' + y = sec(θ) tan(θ) can be found by assuming a solution of the form yc(θ) = Ae^(kθ) + Be^(-kθ), where A and B are constants and k is a real number. Substituting this assumed solution back into the differential equation, we get k^2Ae^(kθ) + k^2Be^(-kθ) + Ae^(kθ) + Be^(-kθ) = sec(θ) tan(θ).

General Solution:

To find the general solution, we need to find the particular solution of the differential equation. In this case, it involves solving for the particular solution using variation of parameters. This technique involves assuming a particular solution of the form yp(θ) = u(θ)(Ae^(kθ) + Be^(-kθ)), where u(θ) is a function to be determined. The particular solution can then be found by substituting this assumed solution back into the differential equation and solving for u(θ) using the method of undetermined coefficients. Once the particular solution is found, the general solution is given by y(θ) = yc(θ) + yp(θ).

Solving by Variation of Parameters:

To solve the differential equation y'' + y = sin^2(x) by variation of parameters, we first find the complementary function and the general solution as described above. We then substitute the given function sin^2(x) for the right-hand side of the differential equation and solve for the unknown parameters in the particular solution using the method of undetermined coefficients. Finally, we combine the complementary function and particular solution to obtain the general solution of the differential equation.