NonHomogeneous Equations and Undetermined Coefficients

[hard_assteel]

NonHomogeneous Equations and Undetermined Coefficients

Find the particular solution;

y''-10y'+25y=-18e^(5t)

here is my work

yp(x)=-Ae^(5t)
yp'(x)=-5Ae^(5t)
yp''(x)=-25Ae^(5t)

plug into equation

[-25Ae^(5t)]-10[-5Ae^(5t)]+25[-Ae^(5t)...

Now; I have 0=-18e^(5t) which doesn't make sense.

Thank You

 

[Mark44]

Why did you use -Ae^(5t) as your particular solution? That is, with the negative sign?

The particular solution you chose happens to be a solution to the homogeneous equation y''-10y'+25y = 0.

The characteristic equation for the homogeneous equation is r^2 - 10r + 25 = 0, and there is a repeated root for r = 5. This means that in addition to Ae^(5t), Bte^(5t) will also be a solution to the homogeneous equation.

For your equation, I would try y = Ct²e^5t as a particular solution.