52 Terminology concerning solutions

The terminology of MATH115 Differential Equations reappears. Consider the inhomogeneous differential equation

$$\frac{dX}{dt}=AX+BU(t).$$

i) The expression $X(t)=\exp (tA)X_0$ with $X_0$ arbitrary is known as the complementary function, since it satisfies the homogeneous differential equation

$$\frac{dX}{dt}=AX.$$

ii) The term ${\int }_0^t\exp ((t-s)A)BU(s)𝑑s$ is a particular integral of the inhomogeneous differential equation.

iii) The general solution of the inhomogeneous equation is a particular integral plus the complementary function.