# What is the general solution of the differential equation? : # (d^2y)/dx^2-dy/dx-2y=4x^2 #

# y = Ae^(-x)+Be^(2x) -2x^2+2x-3#

There are two major steps to solving Second Order DE's of this form:

- Find the Complementary Function (CF) This means find the general solution of the Homogeneous Equation

To do this we look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.

And so the solution to the DE is;

-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Verification:

- Find a Particular Integral* (PI)

This means we need to find a specific solution (that is not already part of the solution to the Homogeneous Equation). As the RHS is a quadratic we try a solution of the quadratic form:

If we substitute into the initial DE we get:

Equating Coefficients we have

So we have found that a Particular Solution is:

- General Solution (GS) The General Solution to the DE is then:

GS = CF + PI

Hence The General Solution to the initial DE is

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The general solution of the given differential equation ( \frac{{d^2y}}{{dx^2}} - \frac{{dy}}{{dx}} - 2y = 4x^2 ) is:

[ y(x) = c_1 e^{2x} + c_2 e^{-x} - x^2 - 2x - 2 ]

where ( c_1 ) and ( c_2 ) are arbitrary constants.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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