Find the solution of the second-order linear differential equation that satisfies the given initial conditions. y'' + 6y' + 9y = 0 y(0) = 0 y'(0) = 1

The solution of this differential system is .

Step-by-step explanation:

This second-order differential equation is homogeneous and linear of the form:

(1)

Where:

- First-order constant coefficient, dimensionless.

- Zero-order constant coefficient, dimensionless.

Whose characteristic polynomial is:

(2)

Where contains the roots associated with the solution of the differential equation.

If we know that and , the roots of the characteristic equation are, respectively:

Which means that and the solution of the differential equation is of the form:

(3)

Where and are integration constants.

The first derivative of the equation above is:

(4)

Now, if we get that and , then the system of equations to the solved is:

(3b)

(4b)

The solution of this system is: , . Therefore, the solution of this differential system is .

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step-by-step explanation: