To solve the differential equation of the form yy’ = sec(x) using the method of variation of parameters, we first need to rewrite the equation in a standard form.
1. **Rewrite the equation**: We can express this as y’ = rac{sec(x)}{y}.
2. **Identify the solution**: Next, we can look for a particular solution. Given this is a non-linear ordinary differential equation, we start with a simpler function for y. A common approach is to assume a solution of the form y = y_h + y_p, where y_h is the homogeneous part and y_p is a particular solution.
3. **Solve the homogeneous equation**: The homogeneous equation associated with this differential equation can be derived by setting the right-hand side to zero. A typical homogeneous solution might take the form y_h = C y_0, where y_0 is a solution of the homogeneous part.
4. **Find the particular solution**: To find y_p, we employ the method of variation of parameters. We systematically calculate the integral of sec(x) in terms of y_h. By substituting y_h into our equation and adjusting for the particular aspect, we can derive an expression for y_p.
5. **Combine the solutions**: After finding both solutions, you can express the final answer as y = y_h + y_p. This gives us the general solution to the original differential equation.
6. **Check your work**: After arriving at the solution, it’s a good idea to differentiate and substitute back into the original differential equation to ensure it holds true.