5.8 Solution of Single State Equations

Let us consider the state equations

where ?, ?, k ₁, and k ₂ are scalar constants, and the initial condition, if non-zero, is denoted as

We will now prove that the solution of the first state equation in (5.111) is

Proof:

First, we must show that (5.113) satisfies the initial condition of (5.112). This is done by substitution of t = t ₀ in (5.113). Then,

The first term in the right side of (114) reduces to x ₀ since

The second term of (5.114) is zero since the upper and lower limits of integration are the same. Therefore, (5.114) reduces to x( t ₀) = x ₀ and thus the initial condition is satisfied.

Next, we must prove that (5.113) satisfies also the first equation in (5.111). To prove this, we differentiate (5.113) with respect to t and we get

We observe that the bracketed terms of (5.116) are the same as the right side of the assumed solution of (5.113). Therefore,

and this is the same as the first equation of (5.111). The second equation of (5.111) is an algebraic equation whose coefficients are scalar constants.

In summary, if ? and ? are scalar constants, the solution of

with initial condition

is obtained from the relation