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# Find the general solution of the given higher-order differential equation. y'''-6y''-7y' = 0

**Solution:**

The characteristic equation of the given differential equation is as follows:

r^{3} - 6r^{2} - 7r = 0

Taking r common we have

r(r^{2} - 6r -7) = 0

Factorising (r^{2} - 6r -7) we get

r(r - 7)(r + 1) = 0

And so the characteristic equation has three distinct real roots r = 0, r = -1, and r = 7.

The general solution can be written as :

y(x) = \( c_{1}e^{0.x} + c_{2}e^{-1x}+ c_{2}e^{7x} \)

Now we know that e^{0} =1

y(x) = \( c_{1} + c_{2}e^{-1x}+ c_{2}e^{7x} \)

The above equation is the general solution of the given equation.

## Find the general solution of the given higher-order differential equation. y'''-6y''-7y' = 0

**Summary:**

The general solution to the third order linear differential equation is : y(x) = \( c_{1} + c_{2}e^{-1x}+ c_{2}e^{7x} \)

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