Consider the equation ay'' + by' + cy = d, where a, b, c, and d are constants. (a) find all equilibrium, or constant, solutions

Question

aalyn [17]

3 years ago

13

Consider the equation ay'' + by' + cy = d, where a, b, c, and d are constants. (a) find all equilibrium, or constant, solutions

of this differential equation. (enter your answers as a comma-separated list of equations.)

Mathematics

1 answer:

Orlov [11] · Answer 1 · 2022-04-07T19:38:21+03:00

Orlov [11]3 years ago

4 0

Rewrite equation so that it is homogeneous .
$ay'' +by' + cy = 0$

Solve characteristic equation:
$ar^2 +br + c = 0 \\ \\ r = \frac{(-b) \pm \sqrt{b^2 -4ac}}{2a}$

The solutions to the homogeneous equation are:
$y = k_1 e^{r_1 t} + k_2 e^{r_2 t}$

Finally you need the constant term in "cy" to equal 'd' to satisfy the particular solution.
$y= k_1 e^{r_1 t} + k_2 e^{r_2 t} + \frac{d}{c}$