Nonhomogeneous Linear Equations

In the previous atom, we learned that a second-order linear differential equation has the form:

$\displaystyle{\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)}$

where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions. When $f(t)=0$, the equations are called homogeneous second-order linear differential equations. Otherwise, the equations are called nonhomogeneous equations. Examples of homogeneous or nonhomogeneous second-order linear differential equation can be found in many different disciplines such as physics, economics, and engineering.

Heat Transfer

Phenomena such as heat transfer can be described using nonhomogeneous second-order linear differential equations.

In simple cases, for example, where the coefficients $A_1(t)$ and $A_2(t)$ are constants, the equation can be analytically solved. (Either the method of undetermined coefficients or the method of variation of parameters can be adopted.) In general, the solution of the differential equation can only be obtained numerically. However, there is a very important property of the linear differential equation, which can be useful in finding solutions.

Linearity

Linear differential equations are differential equations that have solutions which can be added together to form other solutions. If $y_1(t)$ and $y_2(t)$ are both solutions of the second-order linear differential equation provided above and replicated here:

$\displaystyle{\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)}$

then any arbitrary linear combination of $y_1(t)$ and $y_2(t)$ —that is, $y(x) = c_1y_1(t) + c_2 y_2(t)$ for constants $c_1$ and $c_2$—is also a solution of that differential equation. This can be confirmed by substituting $y(x) = c_1y_1(t) + c_2 y_2(t)$ into the equation and using the fact that both $y_1(t)$ and $y_2(t)$ are solutions of the equation.