Variation of Parameters, system of equations

[Linday12]

Homework Statement

y''+25y=cot(5x)
Find one possible solution

The Attempt at a Solution

I don't have any background in linear algebra, so I can't use cramers rule as a heads up, so I have to solve the system of equations (no linear algebra for this course is needed).

Ok, so I take the auxiliary, r^2=-25, r=+/-5i

y_h=Acos(5x)+Bsin(5x)

y_1=cos(5x) y_2=sin(5x)

Then,
u_1'cos(5x)+u_2'sin(5x) = 0
-5u_1'sin(5x)+5u_2'cos(5x) = cot(5x)

I am stumped at how to solve this system of equations. I think the rest before it is right..?

Seems like the algebra has got me.

 

[Linday12]

Perhaps I'm going about this completely wrong then. How would I start out solving y''+25y=cot(5x).

(I should be more specfic, when I say solve, I mean to find A solution, so y_h and y_p. I'm just not sure about how to find the y_p=u1(y_1)+u2(y_2))

 

[rock.freak667]

Ok, so I take the auxiliary, r^2=-25, r=+/-5i

y_h=Acos(5x)+Bsin(5x)

y_1=cos(5x) y_2=sin(5x)

Then,
u_1'cos(5x)+u_2'sin(5x) = 0
-5u_1'sin(5x)+5u_2'cos(5x) = cot(5x)

I am stumped at how to solve this system of equations. I think the rest before it is right..?

Seems like the algebra has got me.

u₁' = -u₂'tan(5x), sub that into the other equation.

this may help you as well sin²x+cos²x=1

Though, you could just use the direct result to make this easier. You are sure you can't use the Wronskian?