Question

Solve the given system of differential equations by systematic elimination. Dx/dt = 6x 10y Dy/dt = x − 3y

(x(t), y(t)) =

Answer 1

To solve the given system of differential equations by systematic elimination, we can use the method of substitution. Isolate one variable in one of the equations, substitute it back into the other equation, and solve for the remaining variable. Substitute this value back into the first equation to find the corresponding value for the other variable.

Step-by-step explanation:

To solve the given system of differential equations by systematic elimination, we can use the method of substitution.

1. First, isolate one variable in one of the equations. Let's isolate x in the first equation: $$x = \frac{{dy}}{{dt}} - 3y$$
2. Next, substitute this expression for x in the second equation: $$\frac{{d(\frac{{dy}}{{dt}} - 3y)}}{{dt}} = 6(\frac{{dy}}{{dt}} - 3y) + 10y$$
3. This is now a first-order linear ordinary differential equation in terms of y. Solve this equation using standard techniques, such as separation of variables.
4. Once you have found the solution for y, substitute it back into the equation from the first step to find the corresponding solution for x.

For example, suppose we find the solution for y to be y = e^2t. Substituting this back into the expression for x, we get x = \frac{{de^2t}}{{dt}} - 3e^2t = 2e^2t - 3e^2t = -e^2t.