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

Answer 1

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

We have:

# y''''-3y'''+3y''-y' = 0 # ..... [A]
This is a Fourth order linear Homogeneous Differentiation Equation with constant coefficients. The standard approach is to find a solution, #y_c# of the homogeneous equation by looking at the Auxiliary Equation, which is the polynomial equation with the coefficients of the derivatives.

Complementary Function

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

# m^4 -3m^3+3m^2-m = 0 #
The challenge with higher order Differential Equation is solving the associated higher order Auxiliary equation. We note that #m# is a factor, so:
# m(m^3 -3m^2+3m-1) = 0 #
By inspection we see #m=1# is a solution of the cubic, this by the factor theorem #m-1# is a factor of the cubic, so we can perform algebraic long division to get:
# m(m-1)^3 = 0 #
Which has three real repeated roots #m=1# and a unique root #m=0#

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:

# y = (Ax^2+Bx+C)e^(1x) + De(0x) # # \ \ = (Ax^2+Bx+C)e^x + D #
Note this solution has #4# constants of integration and #4# linearly independent solutions, hence their superposition is the General Solution.
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Answer 2

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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Answer from HIX Tutor

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.

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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