Differential Equation: u'' + u = tan x

[mario99]

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]

 

[khansaheb]

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]

For (b)

if c_p(x) = \(\displaystyle \int_{}^{x} g(x,s) f(s) \ ds\) can you calculate c' and c". What do you get?

 

[mario99]

For (b)

if c_p(x) = \(\displaystyle \int_{}^{x} g(x,s) f(s) \ ds\) can you calculate c' and c". What do you get?

Thank you khansaheb


[imath] \displaystyle c'(x)_p = g(x) f(x) [/imath]
[imath] \displaystyle c''(x)_p = g'(x) f(x) + g(x)f'(x)[/imath]


How does this help me?

 

[khansaheb]

Now use the original DE to help you further.......

 

[mario99]

Now use the original DE to help you further.......

[imath]u'' + u = \tan x[/imath]

[imath] \displaystyle g'(x) f(x) + g(x)f'(x) + \int_{}^{x} g(x,s) f(s) \ ds = \tan x[/imath]

[imath] \displaystyle g'(x) \tan x + g(x)\sec^2 x + \int_{}^{x} g(x,s) \tan s \ ds = \tan x[/imath]

The functions [imath]g(x) \ \text{and} \ g(x,s)[/imath] are still unknown.