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The equation to be solved now is of the form
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where again  is the first derivative of y(t) and b is independent of the function y(t) and its derivatives.
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To solve for y(t), one may first divide both sides of the equation by
which results in:
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Noting the identity
with
, one can realize that once again the above equation is in the form defined in Case 1.
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Therefore, by integrating the two sides of the above equation one can find:
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That is, one can write:
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where c = y(t0)
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Multiplying both sides of the above by
, the solution can now be written as:
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or equivalently:
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where c = y(t0). Here, since both a and b are constants, the solution can be further simplified as:
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Again, it should be added that this result holds even for first order linear time varying (LTV) differential equations. That is, given the first order differential equation:
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The solution is given as:
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Examples:
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1.
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2.
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3.
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First order non-homogeneous LTI Differential Equations