Variation of Parameters

To solve the second-order linear differential equation d^2y/dx^2 = tan(x), we can use the method of variation of parameters. This method is used when the equation is non-homogeneous and allows us to find a particular solution.

The Homogeneous Equation

First, let's find the general solution to the homogeneous equation d^2y/dx^2 = 0. This equation has the form of a second-order constant coefficient homogeneous equation, and its general solution is given by y_h(x) = C1 + C2x, where C1 and C2 are constants.

Finding the Particular Solution

To find a particular solution, we assume that the particular solution has the form y_p(x) = u1(x)y1(x) + u2(x)y2(x), where y1(x) and y2(x) are the fundamental set of solutions to the homogeneous equation, and u1(x) and u2(x) are functions to be determined.

We can find y1(x) and y2(x) by solving the homogeneous equation. Since the homogeneous equation is d^2y/dx^2 = 0, we obtain y1(x) = 1 and y2(x) = x.

Determining the Functions u1(x) and u2(x)

To determine the functions u1(x) and u2(x), we substitute the particular solution y_p(x) = u1(x)y1(x) + u2(x)y2(x) into the original non-homogeneous equation d^2y/dx^2 = tan(x). We then equate the coefficients of y1(x) and y2(x) to zero.

For y1(x), we have:
d^2(u1(x)y1(x) + u2(x)y2(x))/dx^2 = tan(x)
0 + 0 = tan(x)
This results in the equation 0 = tan(x), which is impossible. Therefore, we equate the coefficient of y1(x) to zero, resulting in:
u1'(x) + x*u2'(x) = 0

For y2(x), we have:
d^2(u1(x)y1(x) + u2(x)y2(x))/dx^2 = tan(x)
0 + 0 = tan(x)
This results in the equation 0 = tan(x), which is impossible. Therefore, we equate the coefficient of y2(x) to zero, resulting in:
u1'(x) + u2(x) = tan(x)

Solving the System of Equations

We now have a system of equations:
u1'(x) + x*u2'(x) = 0
u1'(x) + u2(x) = tan(x)

To solve this system, we can differentiate the first equation with respect to x and substitute the result into the second equation:

u1''(x) + x*u2''(x) + u2'(x) = tan(x)

Since the first term in this equation is zero (from the first equation in the system), we have:

x*u2''(x) + u2'(x) = tan(x)

This is a first-order linear ordinary differential equation for u2(x). Solving this equation will give us the function u2