Final answer:
To show that y = sec(x) is a solution to the differential equation y'' - 2tan(x) y' - y = 0, we differentiate y twice and substitute them into the equation. For the reduction of order, we assume a second solution of the form y = v(x)y₁(x) and solve the resulting second-order homogeneous equation to find a second, linearly independent solution. Finally, we can solve the given initial value problem by using the initial conditions.
Step-by-step explanation:
Mathematics: College
To show that y = sec(x) is a solution to the differential equation y’’ – 2tan(x) y’ – y = 0, we first differentiate y with respect to x twice and substitute them into the equation.
After simplifying the expression, we get that sec(x) satisfies the differential equation. For reduction of order, we assume a second solution of the form y = v(x)y₁(x), where y₁(x) is the known solution of the equation.
By substituting this into the differential equation and simplifying, we obtain a second-order homogeneous equation for v(x). Solving this equation will give us the second, linearly independent solution.
Finally, using the given initial conditions y(0) = 1 and y’(0) = 1, we can determine the specific solution to the initial value problem.