Please help with DE system

[Kotonaru]

I need to solve system of differential equations using elimination method

y1'=y2
y2''=-y1+1/cosx

What I did:

y1'''=y2''

y1'''+y1=1/cosx

Could you say what am I supposed to do next, how to get the answer?

 

[Deleted member 4993]

I need to solve system of differential equations using elimination method

y1'=y2
y2''=-y1+1/cosx

What I did:

y1'''=y2''

y1'''+y1=1/cosx

Could you say what am I supposed to do next, how to get the answer?

Next you have to find the solution for the homogeneous equation:

y₁"' +y₁ = 0

The characteristic equation for the above ODE is:

r³ + 1 = 0 → (r +1)(r² - r + 1) = 0 → r₁ = -1 and r_2,3 = 1/2 ± i\(\displaystyle \dfrac{\sqrt{3}}{2}\)

Now what....

Solution of the particular equation (y_p) will involve hyper-geometric functions (so says wolframalfa.....)

 

[Kotonaru]

I still can't figure out how to get y1,
could you give some more info?

 

[HallsofIvy]

What have you tried? I would recommend the "variation of parameters" method. Do you know how to do that? Do you know what the general solution to the homogeneous equation is? (Subhotosh Khan pretty much gave you that.)