Third Order Homogeneous Linear Differential Equation

[yaarpatandar]

Give an example of a third order homogeneous linear differential equation such that the functions y1=2, y2=e-x, and y3=e2x are its solutions. Justify your claim.


I have no idea on how to do this.

 

[Deleted member 4993]

Give an example of a third order homogeneous linear differential equation such that the functions y1=2, y2=e^-x, and y3=e^2x are its solutions. Justify your claim.


I have no idea on how to do this.

.

Can you do the following:


Give an example of a second order homogeneous linear differential equation such that the functions y1=e^-x & y2=e^2x are its solutions.

 

[HallsofIvy]

A "third order homogeneous linear differential equation" (with constant coefficients) is of the form ay'''+ by''+ cy'+ dy=0. It has "characteristic equation" $\displaystyle at^3+ br^2+ cr+ d= 0$". If r= a is a real root, then (r- a) is a factor of that polynomial and $\displaystyle Ce^{ax}$ is a solution to the differential equation.

Here you are told that solutions are 2 ($\displaystyle = 2e^{0x}$), $\displaystyle e^{-x}$, and $\displaystyle e^{2x}$ so r= 0, -1, and 2.