# #y'' + 9y = 2x^2 - 5#, How about solution. (#y_h#=?, #y_p=#?)

# y(x) = Acos(3x)+Bsin(3x) + 2/9x^2 -49/91 #

We are looking for a fix for

Complementary Role

The equation homogeneous linked to [A] is

Additionally, the related auxiliary equation is:

Consequently, we have the answers:

Parts of the solution are determined by the auxiliary equation's roots; if these parts are linearly independent, the solutions' superposition forms the complete general solution.

Thus, the homogeneous equation [B] has the following solution:

Specific Resolution

To identify a specific non-homogeneous equation solution:

Thus, it is probably best if we search for a solution of this kind:

When we enter these outcomes into the DE [A], we obtain:

Calculating coefficients gives us:

Thus, we arrive at the specific solution:

Overall Resolution

which ultimately results in the GS of [A}

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The homogeneous solution (y_h) is y_h = C₁*cos(3x) + C₂*sin(3x), where C₁ and C₂ are arbitrary constants.

The particular solution (y_p) is y_p = (1/18)*x^2 - (5/18).

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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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