Question

Variation of Parameters ODE problem, posted below shortly.

Answer 1

\[\text{Solve the differential equation by variation of parameters.}\]\[y''+y=\sin(x)\]

Answer 2

Alright, so the coefficient to the highest term is already constant. I think what I need to do now, if I remember correctly, is set up the Characteristic Equation.

\[m^2+m=0; \ \ \ m(m+1)=0; \ \ \ m=-1,0.\]

Answer 3

Not sure if I remember how I move forward from here.

Answer 4

first the general solution

Answer 5

\[y''+y=0\\
assume~ form~~ y=e^{mx}\\
y''=m^2*e^{mx}\\
plug into main equation
e^{mx}(m^2+1)=0\\
(m^2+1)=0\\
m=+/- i
y=e^(+/-ix)\\
e^ix= cosx+isinx\\
e^-ix=cosx-isinx\\
since~ y=c1e^{ix}+c2e^{ix}\\
you~can~certainly~take~c1~and~c2~to~left~with~only~sinx~and~cosx\\
~if~c1~and~c2~are~real~and~cx\\
y=c1cosx+c2sinx\]
you can also also say cosx and sin x are solution by principal of linear super position

Answer 6

now for the variation of parameters part

Answer 7

instead of assuming the change is only by a factor, we assume a change wrt to a parameter

Answer 8

y=u1(x)sinx+u2cos(x) as a soln of y''+y=sin(x)

Answer 9

sry back i forgot i was doing this problem lol