# What is the general solution of the differential equation # y''''-3y'''+3y''-y' = 0 #?

# y = (Ax^2+Bx+C)e^x + D #

We have:

Complementary Function

The Auxiliary equation associated with the homogeneous equation of [A] is (as indicated)(:

The roots of the auxiliary equation determine parts of the solution, which if linearly independent then the superposition of the solutions form the full general solution.

Thus the solution of the homogeneous equation [A} is of the form:

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The general solution of the differential equation ( y'''' - 3y''' + 3y'' - y' = 0 ) can be found by solving the characteristic equation associated with the differential equation.

The characteristic equation is obtained by substituting ( y = e^{rx} ) into the differential equation, which yields ( r^4 - 3r^3 + 3r^2 - r = 0 ).

Factoring out ( r ), we get ( r(r^3 - 3r^2 + 3r - 1) = 0 ).

This equation can be further factored into ( r(r - 1)^3 = 0 ).

The roots of this equation are ( r = 0 ) (with multiplicity 1) and ( r = 1 ) (with multiplicity 3).

Thus, the general solution of the differential equation is ( y = C_1 + C_2e^x + C_3xe^x + C_4x^2e^x ), where ( C_1, C_2, C_3, ) and ( C_4 ) 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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