Question

Variation of parameters, to solve: y′′−2tan(x)y′−y=sec(x),y(0)=1;y′(0)=1

Answer 1

To solve the given differential equation using variation of parameters, follow the steps: find the complementary solution, find the particular solution, substitute the assumed particular solution, solve the resulting system of equations, and find the general solution.

Step-by-step explanation:

To solve the given differential equation using variation of parameters, we need to follow these steps:

1. Find the complementary solution by solving the associated homogeneous equation y'' - 2tan(x)y' - y = 0.
2. Find the particular solution by assuming the form y_p = u(x)y₁(x) + v(x)y₂(x), where y₁(x) and y₂(x) are linearly independent solutions of the homogeneous equation, and u(x) and v(x) are functions to be determined.
3. Substitute the assumed particular solution into the original differential equation and equate the coefficients of the linearly independent solutions to zero.
4. Solve the resulting system of equations to find the functions u(x) and v(x).
5. The general solution is given by y = y_h + y_p, where y_h is the complementary solution and y_p is the particular solution.

By following these steps, you can find the solution to the given differential equation.