Question

Solve x(t) for 2x”(t)+6x’(t)+4x(t)=12, x(0)=3, x’(0)=1.

Answer 1

You are going to have to learn some technique to be able to solve this. What have you been taught so far?

Answer 2

Still there? Can't help if you don't talk to me.

Answer 3

how do u rewrite the expression for 2x''(t)

Answer 4

That's not the approach.

What technique or techniques have you been taught for equations like this?

Answer 5

well, i need to write the eq into standard form

6x'(t)= 6dx/dt
what is 2x''(t)????

Answer 6

x'' is the second derivative of the function x(t) with respect to t.

Answer 7

In short the approach is this
- solve the homogeneous equation
2x'' + 6x' + 4x = 0
* To do this, substitute x(t) = e^(rt) and find a quadratic equation in r
* solve the equation to find values of r, call them r1 and r2
* then the general solution of the homogeneous equation is
\[ x_h(t) = c_1 e^{r_1t} + c_2 e^{r_2t} \]

- now find a particular solution to the inhomogeneous equation
2x'' + 6x' + 4x = 12
* I'll tell you what it is: \( x_p(t) = 3 \). Substitute it in and you'll see it works.

- the general solution of the inhomogeneous equation is the sum of the homogeneous and particular solutions:
\[ x(t) = x_h(t) + x_p(t) = c_1 e^{r_1t} + c_2 e^{r_2t} + 3 \]

- finally, apply the initial conditions of
\[ x(0)=3, \ \ x’(0)=1 \]
to that general solution to solve for the constants \( c_1 \) and \( c_2 \) .