11.2: B- Integration of Some Basic Linear ODEs

In this appendix we collect together a few common ideas related to solving, explicitly, linear inhomogeneous differential equations. Our discussion is organized around a series of examples.

Example \(\PageIndex{32}\)

Consider the one dimensional, autonomous linear vector field:

\[\dot{x} = ax, x,a \in \mathbb{R}. \label{B.1}\]

We often solve problems in mathematics by transforming them into simpler problems that we already know how to solve. Towards this end, we introduce the following (time-dependent) transformation of variables:

\[x = ue^{at}. \label{B.2}\]

Differentiating this expression with respect to t, and using (B.1), gives the following ODE for u:

\[\dot{u} = 0, \label{B.3}\]

which is trivial to integrate, and gives:

\[u(t) = u(0), \label{B.4}\]

and it is easy to see from (36) that:

\[u(0) = x(0). \label{B.5}\]

Using (36), as well as (B.4) and (B.5), it follows that:

\[x(t)e^{at} = u(t) = u(0) = x(0), \label{B.6}\]

or

\[x(t) = x(0)e^{at}. \label{B.7}\]

Example \(\PageIndex{33}\)

Consider the following linear inhomogeneous nonautonomous ODE (due to the presence of the term b(t)):

\[\dot{x} = ax+b(t), a, x \in \mathbb{R}, \label{B.8}\]

where b(t) is a scalar valued function of t, whose precise properties we will consider a bit later. We will use exactly the same strategy and change of coordinates as in the previous example:

\[x = ue^{at}. \label{B.9}\]

Differentiating this expression with respect to t, and using (B.8), gives:

\[\dot{u} = e^{-at}b(t). \label{B.10}\]

(Compare with (B.3).) Integrating (B.10) gives:

\[u(t) = u(0) + \int_{0}^{t} e^{-at'} b(t')dt'. \label{B.11}\]

Now using (B.9) (with the consequence u(0) = x(0)) with (B.11) gives:

\[x(t) = x(0)e^{at} + e^{at} \int_{0}^{t} e^{-at'} b(t')dt'. \label{B.12}\]

Finally, we return to the necessary properties of b(t) in order for this unique solution of (B.8) to "make sense". Upon inspection of (B.12) it is clear that all that is required is for the integrals involving b(t) to be well-defined. Continuity is a sufficient condition.

Example \(\PageIndex{34}\)

Consider the one dimensional, nonautonomous linear vector field:

\[\dot{x} = a(t)x, x \in \mathbb{R}, \label{B.13}\]

where a(t) is a scalar valued function of t whose precise properties will be considered later. The similarity between (B.1) and (B.13) should be evident. We introduce the following (time-dependent) transformation of variables (compare with (36)):

\[x = ue^{\int_{0}^{t} a(t')dt'}. \label{B.14}\]

Differentiating this expression with respect to t, and substituting (B.13) into the result gives:

\(\dot{x} = \dot{u}e^{\int_{0}^{t} a(t')dt'}+ua(t)e^{\int_{0}^{t} a(t')dt'}\),

\[= \dot{u}e^{\int_{0}^{t} a(t')dt'}+a(t)x, \label{B.15}\]

which reduces to:

\[\dot{u} = 0. \label{B.16}\]

Integrating this expression, and using (B.14), gives:

\[u(t) = u(0) = x(0) = x(t)e^{-\int_{0}^{t} a(t')dt'}, \label{B.17}\]

or

\[x(t) = x(0)e^{\int_{0}^{t} a(t')dt'}. \label{B.18}\]

As in the previous example, all that is required for the solution to be well-defined is for the integrals involving a(t) to exist. Continuity is a sufficient condition.

Example \(\PageIndex{35}\)

Consider the one dimensional inhomogeneous nonautonomous linear vector field:

\[\dot{x} = a(t)x + b(t), x \in \mathbb{R}, \label{B.19}\]

where a(t), b(t) are scalar valued functions whose required properties will be considered at the end of this example. We make the same transformation as (B.14):

\[x = ue^{\int_{0}^{t} a(t')dt'}, \label{B.20}\]

from which we obtain:

\[\dot{u} = b(t)e^{-\int_{0}^{t} a(t')dt'}. \label{B.21}\]

Integrating this expression gives:

\[u(t) = u(0)+ \int_{0}^{t} b(t')e^{-\int_{0}^{t'} a(t'')dt''}dt'. \label{B.22}\]

Using gives:

\[x(t)e^{-\int_{0}^{t} a(t')dt'} = x(0)+ \int_{0}^{t} b(t')e^{-\int_{0}^{t'} a(t'')dt''}dt'. \label{B.23}\]

or

\[x(t) = x(0)e^{\int_{0}^{t} a(t')dt'}+e^{\int_{0}^{t} a(t')dt'} \int_{0}^{t} b(t')e^{-\int_{0}^{t'} a(t'')dt''}dt'. \label{B.24}\]

As in the previous examples, that all that is required is for the integrals involving a(t) and b(t) to be well-defined. Continuity is a sufficient condition.

The previous examples were all one dimensional. Now we will consider two n dimensional examples.

Example \(\PageIndex{36}\)

Consider the n dimensional autonomous linear vector field:

\[\dot{x} = Ax, x \in \mathbb{R}^n, \label{B.25}\]

where A is a \(n \times n\) matrix of real numbers. We make the following transformation of variables (compare with):

\[x = e^{At}u. \label{B.26}\]

Differentiating this expression with respect to t, and using (B.25), gives:

\[\dot{u} = 0. \label{B.27}\]

Integrating this expression gives:

\[u(t) = u(0). \label{B.28}\]

Using (B.26) with (B.28) gives:

\[u(t) = e^{-At}x(t) = u(0) = x(0), \label{B.29}\]

from which it follows that:

\[x(t) = e^{At} x(0). \label{b.30}\]

Example \(\PageIndex{37}\)

Consider the n dimensional inhomogeneous nonautonomous linear vector field:

\[\dot{x} = Ax+g(t), x \in \mathbb{R}^n, \label{B.31}\]

where g(t) is a vector valued function of t whose required properties will be considered later on. We use the same transformation as in the previous example:

\[x = e^{At}u. \label{B.32}\]

Differentiating this expression with respect to t, and using (B.31), gives:

\[\dot{u} = e^{At}g(t), \label{B.33}\]

from which it follows that:

\[u(t) = u(0)+ \int_{0}^{t} e^{-At'}g(t')dt', \label{B.34}\]

or, using (B.32)

\[x(t) = e^{At}x(0) + e^{At} \int_{0}^{t} e^{-At'}g(t')dt'. \label{B.35}\]