verify that y=C1e^2x + C2e^-3 is a solution of the differential equation y''+y'-6y=0

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To verify that y = C1e^2x + C2e^-3 is a solution of the differential equation y''+y'-6y = 0, we need to substitute y into the equation and check if it satisfies the equation.

First, let's find the first and second derivative of y:
y' = d/dx(C1e^2x + C2e^-3) = 2C1e^2x - 3C2e^-3
y'' = d^2/dx^2(C1e^2x + C2e^-3) = 4C1e^2x + 9C2e^-3

Now let's substitute these derivatives into the differential equation:
y'' + y' - 6y = (4C1e^2x + 9C2e^-3) + (2C1e^2x - 3C2e^-3) - 6(C1e^2x + C2e^-3)
= 6C1e^2x + 6C2e^-3 - 6C1e^2x - 6C2e^-3
= 0

As we can see, when we substitute y into the differential equation, we obtain 0. Therefore, y = C1e^2x + C2e^-3 is indeed a solution of the differential equation y'' + y' - 6y = 0.

To verify if y = C1e^2x + C2e^-3x is a solution of the differential equation y'' + y' - 6y = 0, we need to substitute it into the equation and see if it satisfies the differential equation.

Step 1: Compute the first and second derivatives of y with respect to x.

The first derivative of y with respect to x (y') is given by:
y' = d/dx (C1e^2x + C2e^-3x)
= C1(2e^2x) + C2(-3e^-3x)
= 2C1e^2x - 3C2e^-3x

The second derivative of y with respect to x (y'') is given by:
y'' = d/dx (2C1e^2x - 3C2e^-3x)
= 2C1(2e^2x) + 3C2(3e^-3x)
= 4C1e^2x + 9C2e^-3x

Step 2: Substitute the expressions for y, y', and y'' into the differential equation.

Substituting y = C1e^2x + C2e^-3x, y' = 2C1e^2x - 3C2e^-3x, and y'' = 4C1e^2x + 9C2e^-3x into the differential equation y'' + y' - 6y = 0, we get:

(4C1e^2x + 9C2e^-3x) + (2C1e^2x - 3C2e^-3x) - 6(C1e^2x + C2e^-3x) = 0

Simplifying the equation, we have:

4C1e^2x + 9C2e^-3x + 2C1e^2x - 3C2e^-3x - 6C1e^2x - 6C2e^-3x = 0

Now, we can collect like terms:

(4C1 + 2C1 - 6C1)e^2x + (9C2 - 3C2 - 6C2)e^-3x = 0

Combining the coefficients:

0e^2x + 0e^-3x = 0

0 + 0 = 0

Since 0 = 0, the equation is satisfied.

Conclusion:
After substituting y = C1e^2x + C2e^-3x into the differential equation y'' + y' - 6y = 0, we found that the equation is satisfied. Therefore, y = C1e^2x + C2e^-3x is indeed a solution of the given differential equation.