Is my solution for a differential equation with a forcing function correct?

[zoom1]

Have an equation;

d²y(t)/dt² + 5d²y(t)/dt² + 4y(t) = 2e^-2t

Solved the complementary(homogenous) part and the function and got the roots of -1 and -4

so the y_h(t) is A₁.e^-4t + A₂.e^-t

Forcing function is 2.e^-2t so y_particular(t) is A.e^-2t

Am I right here ? Or am I supposed to use Ate^-2t

Well, if I use the first one, the resultant function doesn't give me the 2.e^-2t when I put it into the differential equation, so there is something wrong obviously.

However F(t) or one of its derivatives are not identical to terms in the homogenous solution, so I think I have to use the first option, which is A.e^-2t

After proceeding I ended up with y_p(t) = 1/3.e^-2t

Initial values are y(0) = 0 and y⁽¹⁾(0) = 0

so, K₁ = -1/9 and K₂ = -2/9

Still couldn't find where I am wrong
Appreciate if you help me.

 

[gulfcoastfella]

Have an equation;

d²y(t)/dt² + 5d²y(t)/dt² + 4y(t) = 2e^-2t

Solved the complementary(homogenous) part and the function and got the roots of -1 and -4

For the differential equation you posted, the above roots are not correct. Did you intend for there to be two second order terms in the equation when you posted it?

 

[zoom1]

For the differential equation you posted, the above roots are not correct. Did you intend for there to be two second order terms in the equation when you posted it?

Ohh, pardon me, the second term is not the second derivative, it had to be first derivative.

 

[Office_Shredder]

You should double check your calculation for Ae^-2t

 

[zoom1]

You should double check your calculation for Ae^-2t

Got it! Thank you ;)

 

[gulfcoastfella]

I got a particular solution of y_{p} = -e^{-2t}. See if you get the same.

 

[dash-dot]

I get A_1=\frac{1}{3} and A_2=\frac{3}{2}. The particular solution given above is correct.

Double-check your calculations.