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The equation to be solved is now considered to be of the form:
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derivative of y(t) and that ak’s
and b are independent of the function y(t) and its derivatives.
Also, u(t) is a function of time.
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To solve for y(t), one can first arrange the above nth order
equation in terms of n first order equations, which results in a vector of
first order LTI differential equation.
Then the results for the first order scalar equations can be extended to the first order vector equations.
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To do this, let us define the following variables:
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Then the first derivative of these variables can be found as:
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where the last line of the above equation was found by substituting for
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from the original nth order differential equation defined at the beginning of this section.
Now denoting the vector x(t)
and its first derivative
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as:
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and
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the above set of n first order
differential equations can be written in a compact vectorized
form, with y(t) = x1(t), as:
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where
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and 
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It can be seen that the above differential equation for x(t) is in the form of non-homogeneous first order LTI
differential equation, described in the previous section, with the exception
that it is now in the vector form.However,
our previous results for the scalar case can also be directly applied for the
vector case, which is given as follows:
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where
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and
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where I is the nXn identity matrix.When A and B are a constant matrices, it is
easy to see that
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and also
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where 
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denotes inverse Laplace transformation.
Hence, the solution can be simplified as:
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or equivalently as:
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where
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is known from the initial conditions.
Again, it should be added that this result holds even for the
nth order linear time varying (LTV) differential equations. That is, consider the
nth order linear differential equation:
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Then, one can find an equivalent first order linear differential
equation in vector form as:
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The solution of this differential equation is then given as:
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| or equivalently, as:
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where
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is known from the
initial conditions.
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Example:
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1) 
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Equivalent vector differential equation is:
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where u(t)=1,  ,
and .
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Then, one can find:
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Now, the solution can be found as:
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or equivalently as:
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That is the solution is given as:
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Nth order non-homogeneous LTI differential equations