Solve the differential equation [imath]u'' + u = \tan x[/imath].
The solution will be in the form of [imath]\displaystyle u(x) = u_c(x) + u_p(x) = c_1u_1(x) + c_2u_2(x) + \int_{}^{x} g(x,s) f(s) \ ds[/imath].
where [imath]u_c(x)[/imath] is the complementary solution, [imath]u_p(x)[/imath] is the particular solution, and [imath]g(x,s)[/imath] is the kernel in the integral.
(a) Find [imath]u_c(x)[/imath] and show that [imath]u_1(x)[/imath] and [imath]u_2(x)[/imath] are linearly independent.
(b) Find [imath]u_p(x)[/imath]. You will need part (a) to solve part (b).
(c) Explain briefly why the differential equation [imath]5u'' + u = \tan x[/imath] is very difficult to solve and when it is written as [imath]\displaystyle 5u'' + u = \tan(x/\sqrt{5})[/imath], it can be solved easily by the technique above.
I could solve (a) easily. I am having trouble in (b) and (c). I don't know how to find [imath]g(x,s).[/imath]
The solution will be in the form of [imath]\displaystyle u(x) = u_c(x) + u_p(x) = c_1u_1(x) + c_2u_2(x) + \int_{}^{x} g(x,s) f(s) \ ds[/imath].
where [imath]u_c(x)[/imath] is the complementary solution, [imath]u_p(x)[/imath] is the particular solution, and [imath]g(x,s)[/imath] is the kernel in the integral.
(a) Find [imath]u_c(x)[/imath] and show that [imath]u_1(x)[/imath] and [imath]u_2(x)[/imath] are linearly independent.
(b) Find [imath]u_p(x)[/imath]. You will need part (a) to solve part (b).
(c) Explain briefly why the differential equation [imath]5u'' + u = \tan x[/imath] is very difficult to solve and when it is written as [imath]\displaystyle 5u'' + u = \tan(x/\sqrt{5})[/imath], it can be solved easily by the technique above.
I could solve (a) easily. I am having trouble in (b) and (c). I don't know how to find [imath]g(x,s).[/imath]