Exercise 0.2.9 : verify that x = c 1 e − t + c 2 e 2 t is a solution to x ′′ − x ′ − 2 x = 0 . find c 1 and c 2 to solve for the initial conditions x (0) = 10 and x ′ (0) = 0 .

Since 2 and -1 are eigenvalues of the differential equation,

is a solution to the differential equation

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The solution to the initial value problem is:

Step-by-step explanation:

We have the following differential equation:

The first step is finding the eigenvalues for this differential equation, that is, finding the roots of the following second order equation:

Since 2 and -1 are eigenvalues of the differential equation,

is a solution to the differential equation.

Solution of the initial value problem:

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So, we have to solve the following system:

The solution to the initial value problem is:

it happens during combustion and impacts the environment because the plants and trees take in that and it can cause them to die.