Are the particular integral and complementary function solutions of a Differential Equation linearly independent?

Answer 1

Yes, Because of the Principle of Superposition

Yes the solutions #y_c# and #y_p# must be linearly independent

Why? Because of the Principle of Superposition

If it is known that the solutions #y_1#, #y_2#.....#y_n#, in #y_c#, are fundamental set of solutions to the homogeneous equation, and are linearly independent then from the Principle of Superposition
#y_("sup") = c_1y_1 + c_2y_2 + ... c_ny_n# where #c_1, c_1, ... c_n# are constants is also a solution of the homogeneous equation.
So then it follows that if #y_c# and #y_p# were not linearly independent then #y_p# could be written as superposition of the existing solutions that form #y_c#.

Therefore it would be a solution of the homogeneous equation, and therefore it would not be the general solution of the non-homogeneous equation

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Answer 2

Yes, the particular integral and complementary function solutions of a linear homogeneous ordinary differential equation are linearly independent. This is a consequence of the principle of superposition, which states that the sum of any two solutions of a linear homogeneous differential equation is also a solution. As a result, the particular integral and complementary function solutions form a linearly independent set.

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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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