Higher-Order Differential Equations

[recon_ind]

Homework Statement

Find the general solution of the given higher-order differential equation.

d³x/(dt³) - d²x/(dt²) - 4x = 0

Homework Equations

Use an auxiliary equation such as m³ - m² - 4 = 0

The Attempt at a Solution

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[Mark44]

That is the auxiliary equation you want to use. As it turns out, the left side can be factored, yielding (m - 2)(m² + m + 2) = 0

There are three distinct solutions, so the solution to this homogeneous problem will be x(t) = c₁e^2t + c₂e^m₂t + c₃e^m₃t. All you have to do is find the other two constants, m₂ and m₃.

 

[recon_ind]

So, I should just use the quadratic formula to find the other two solutions?

Thanks man.

 

[recon_ind]

The other two solutions are of the form $$\alpha$$ $$\pm$$ $$\beta$$

 

[Mark44]

So, I should just use the quadratic formula to find the other two solutions?

Yes. And yes, the two solutions are of the form a +/- b. One of your values of m will be a + b, and the other will be a - b.