Solution of the Differential Equation (D^2y tan(x))

The differential equation given is (D^2y tan(x)), where (D) represents the derivative operator. This equation requires finding both the homogeneous and particular solutions to determine the general solution.

Step 1: Finding the Homogeneous Solution

The associated homogeneous equation is (D^2y - y 0). To solve this, we form the characteristic equation:

[r^2 - 1 0]

Solving the characteristic equation, we get the roots:

[r pm i]

The general solution to the homogeneous equation is:

[y_h C_1 cos(x) C_2 sin(x)]

where (C_1) and (C_2) are constants determined by initial conditions.

Step 2: Finding a Particular Solution

To find a particular solution (y_p) for the non-homogeneous equation, we use the method of variation of parameters. This method is suitable because (tan(x)) does not fit the standard form required for the method of undetermined coefficients.

We first need the Wronskian (W) of the homogeneous solutions (cos(x)) and (sin(x)):

[W begin{vmatrix} cos(x) sin(x) -sin(x) cos(x) end{vmatrix} cos^2(x) - (-sin^2(x)) 1]

The particular solution can be found using the formulas for variation of parameters:

[y_p -int frac{sin(x) tan(x)}{W} , dx int frac{cos(x) tan(x)}{W} , dx]

Substituting the Wronskian value, we get:

[y_p -int sin(x) tan(x) , dx int cos(x) tan(x) , dx]

We can simplify the integrals as follows:

For the first integral:

[-int sin(x) tan(x) , dx -int frac{sin^2(x)}{cos(x)} , dx]

This integral can be solved using substitution or integration by parts.

For the second integral:

[int cos(x) tan(x) , dx int cos(x) frac{sin(x)}{cos(x)} , dx int sin(x) , dx -cos(x)]

Therefore, the particular solution (y_p) is a combination of the solutions to these integrals.

Step 3: Combining the Solutions

The general solution (y) of the differential equation (D^2y tan(x)) is given by combining the homogeneous and particular solutions:

[y y_h y_p C_1 cos(x) C_2 sin(x) y_p]

where (y_p) is the particular solution derived from integrating the expressions involving (tan(x)). The specific integration techniques can be further applied to simplify (y_p).

Summary

The general solution (y) of the equation (D^2y tan(x)) is:

[y C_1 cos(x) C_2 sin(x) y_p]

where (y_p) is the particular solution derived from integrating the expressions involving (tan(x)). This solution can be further simplified with specific integration techniques.

Keywords: Differential Equation, Tan(x), Variation of Parameters