Typically people use long division. But in this case it's obvious that the other factor is r^2+1.hunt_mat said:So what I would do is write:
[tex]
(r-1)(ar^{2}+br+c)=r^{3}-r^{2}+r-1
[/tex]
Expand and equate coefficients, then solve the quadratic
A characteristic equation is an equation that is used to find the roots or solutions of a differential equation. In the case of solving the equation y'''-y''+y'-y=0, the characteristic equation would be r^3-r^2+r-1=0.
To solve a characteristic equation, you would typically use algebraic methods to factor the equation and find the roots. In some cases, you may also need to use the quadratic formula or other methods to find the roots.
Yes, it is possible to solve a characteristic equation using numerical methods such as the Newton-Raphson method or the bisection method. These methods involve using a computer program to approximate the roots of the equation.
Solving the characteristic equation allows us to find the general solution of a differential equation. This is important because it gives us a complete understanding of the behavior of the system described by the equation and allows us to make predictions about its future behavior.
Yes, there are some specific techniques that can be used to solve certain types of characteristic equations. For example, if the equation has repeated roots, we can use the method of reduction of order. If the equation has complex roots, we can use the method of undetermined coefficients. Other techniques include the method of variation of parameters and the method of Laplace transforms.