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Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Second-Order Linear Equations
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Second-Order Linear Equations
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
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Calculus
Concept Version 8
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Nonhomogeneous Linear Equations

Nonhomogeneous second-order linear equation are of the the form: $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $f(t)$ is nonzero.

Learning Objective

  • Identify when a second-order linear differential equation can be solved analytically


Key Points

    • Examples of homogeneous or nonhomogeneous second-order linear differential equation can be found in many different disciplines such as physics, economics, and engineering.
    • In simple cases, for example, where the coefficients $A_1(t)$ and $A_2(t)$ are constants, the equation can be analytically solved. In general, the solution of the differential equation can only be obtained numerically.
    • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.

Term

  • linearity

    a relationship between several quantities which can be considered proportional and expressed in terms of linear algebra; any mathematical property of a relationship, operation, or function that is analogous to such proportionality, satisfying additivity and homogeneity


Full Text

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.

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