DE-BOOK Linearity

Section 1.7 Linearity

In the world of differential equations, distinguishing between linear and nonlinear equations is vital. This distinction will often guide you on how to approach solving the equation and what methods to use. The good news is, all the work we did in the previous section will make this task much easier.

Definition 15.

A differential equation is linear if it contains only linear terms. That is, it can be expressed as

\begin{matrix} (3) & \underset{linear term}{\underset{⏟}{a_{n} (t) y^{(n)}}} + \dots + \underset{linear term}{\underset{⏟}{a_{2} (t) y^{″}}} + \underset{linear term}{\underset{⏟}{a_{1} (t) y^{'}}} + \underset{linear term}{\underset{⏟}{a_{0} (t) y}} = \underset{linear term}{\underset{⏟}{f (t) .}} \end{matrix}

If just one of the terms in the equation is nonlinear, then the entire differential equation is nonlinear.

This definition might seem abstract at first, but it encompasses all possible linear differential equations. In practice, most equations you’ll encounter will involve only a few terms, and the challenge lies in identifying whether any of those terms break the rule of linearity. If they do, the equation is nonlinear.

Example 16. Determine if the equation is linear or nonlinear.

Solution.

y^{″} + \frac{y^{'}}{t^{2}} + y = 17 t

To classify this equation, we need to verify if every term involving

y

and its derivatives is linear. Let’s break it down term by term:

\begin{matrix} y^{″} + \frac{y^{'}}{t^{2}} + y = 17 t \\ \underset{linear}{\underset{―}{y^{″}}} + \underset{linear}{\underset{―}{\frac{1}{t^{2}} y^{'}}} + \underset{linear}{\underset{―}{y}} = \underset{constant term}{\underset{―}{17 t}} \end{matrix}

Since each term involving

y

or its derivatives is linear, this differential equation is indeed linear.

In summary, recognizing whether a differential equation is linear or nonlinear will help you determine the appropriate methods for solving it. Linear equations allow for a more straightforward approach, while nonlinear equations often require specialized techniques.

Reading Questions Check-Point Questions

1. The differential equation $\frac{d y}{d t} + t^{2} y = e^{t}$ is linear.

The differential equation $\frac{d y}{d t} + t^{2} y = e^{t}$ is linear

True
Correct! This equation is linear as all terms involving $y$ and its derivatives appear to the first power, and there are no nonlinear functions of $y .$
False
Incorrect. This equation is indeed linear because both $y$ and $\frac{d y}{d t}$ are to the first power and are not inside any nonlinear functions.

2. The differential equation, $y^{″} + y^{'} \cos t = 7 y,$ is $\underset{―}{}$ .

The differential equation, $y^{″} + y^{'} \cos t = 7 y,$ is

Linear
Yes! This DE is linear. Each term involving $y$ or its derivatives, such as $y^{″},$ $y^{'} \cos t,$ and $7 y,$ is linear. A linear differential equation contains terms where the dependent variable and its derivatives appear to the first power and are not multiplied by each other.
Nonlinear
No, this is linear. Looking carefully at each term, we see:

$\begin{matrix} y^{″} + y^{'} \cos t = 7 y \\ \underset{linear}{\underset{―}{(1) y^{″}}} + \underset{linear}{\underset{―}{(\cos t) y^{'}}} = \underset{linear}{\underset{―}{7 y}} \end{matrix}$

Since every term is linear, this differential equation is linear. Review the definition of linear differential equations.

3. The differential equation, $y^{″} + \sin (y) = 17 t,$ is $\underset{―}{}$ .

The differential equation, $y^{″} + \sin (y) = 17 t,$ is

Linear
No, this is nonlinear. Looking carefully at each term, we see:

$\begin{matrix} y^{″} + \sin (y) = 17 t \\ \underset{linear}{\underset{―}{(1) y^{″}}} + \underset{nonlinear}{\underset{―}{\sin (y)}} = \underset{linear}{\underset{―}{17 t}} \end{matrix}$

Since one term is not linear, this differential equation is nonlinear. Revisit the rules for linearity and nonlinearity.
Nonlinear
Yes! This DE is nonlinear since the $\sin (y)$ term is not linear. Nonlinear terms involve functions like sine, logarithms, or powers greater than one when applied to the dependent variable $y .$

4. The differential equation, $\frac{d^{2} x}{d t^{2}} + e^{x} = 0,$ is $\underset{―}{}$ .

The differential equation, $\frac{d^{2} x}{d t^{2}} + e^{x} = 0,$ is

Linear
Incorrect. The term $e^{x}$ makes this equation nonlinear, as it involves the exponential function of the dependent variable.
Nonlinear
Correct! The term $e^{x}$ introduces nonlinearity into the equation, as it involves the dependent variable $x$ inside an exponential function.

5. Select the linear differential equation.

Select the linear differential equation

$y^{″} + y^{3} = \sin (t)$
Incorrect. The term $y^{3}$ makes this equation nonlinear.
$y^{″} + \cos (y) = 0$
Incorrect. The term $\cos (y)$ makes this equation nonlinear.
$y^{″} + y^{'} + y = 0$
Correct! All terms are linear in this equation, making it a linear differential equation.
$y^{'} + y^{2} = t$
Incorrect. The term $y^{2}$ introduces nonlinearity in this equation.

6. Which term makes the equation $y^{‴} + 3 y^{'} \sin (t) + y^{2} = 0$ nonlinear?

Which term makes the equation $y^{‴} + 3 y^{'} \sin (t) + y^{2} = 0$ nonlinear?

$y^{2}$
Correct! The term $y^{2}$ is nonlinear because the dependent variable $y$ is raised to the second power.
$3 y^{'} \sin (t)$
Incorrect. While this term includes a function of $t,$ it is still linear because $y^{'}$ appears to the first power.
$y^{‴}$
Incorrect. The term $y^{‴}$ is linear because $y$ and its derivatives are to the first power.

7. Click-on all the linear differential equations.

Click-on all the linear differential equations

Linear equations only involve the dependent variable and its derivatives to the first power, and they won’t be inside nonlinear functions like sine or multiplied by each other.

$y^{″} + \sin (y) = 17 t$	$y^{″} + \frac{y^{'}}{t^{2}} + y = 17 t$	$y^{″} + 3 y^{'} + 2 y = 0$
$y^{″} + y^{2} = 17 t$	$y^{″} + \frac{y^{'}}{t} + y = 17 t$	$y = y^{'}$

Hint.

Remember that a linear differential equation contains only linear terms. Four of these equations are linear.

8. Click-on all the nonlinear differential equations.

Click-on all the nonlinear differential equations

Nonlinear equations often have terms where the dependent variable or its derivatives are raised to powers other than one, or are inside functions like sine, logarithms, or are multiplied by each other.

$\frac{d x}{d s} = x^{2} - 4$	$\frac{d^{2} u}{d z^{2}} - 5 \frac{d u}{d z} + 6 u = 0$	$\frac{d p}{d τ} + \sin (p) = τ^{2}$
$\frac{d w}{d v} + 2 v w = \cos (v)$	$\frac{d r}{d θ} + r^{3} = θ$	$\frac{d N}{d t} = - k N$
$\frac{d m}{d q} = m^{3} - q^{2}$	$\frac{d z}{d t} + z \frac{d z}{d t} = t^{3}$	$\frac{d y}{d x} = y \ln (y)$

Hint. Hint

First identify the dependent variable, then carefully look at each term to determine if it is nonlinear.

You have attempted 1 of 9 activities on this page.

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Section 1.7 Linearity

Definition 15.

Example 16. Determine if the equation is linear or nonlinear.

Reading Questions Check-Point Questions

1. The differential equation dydt+t2y=et is linear.

2. The differential equation, ,y″+y′cos⁡t=7y, is ―.

3. The differential equation, ,y″+sin⁡(y)=17t, is ―.

4. The differential equation, ,d2xdt2+ex=0, is ―.

5. Select the linear differential equation.

6. Which term makes the equation y‴+3y′sin⁡(t)+y2=0 nonlinear?

7. Click-on all the linear differential equations.

8. Click-on all the nonlinear differential equations.

1. The differential equation $\frac{d y}{d t} + t^{2} y = e^{t}$ is linear.

2. The differential equation, $y^{″} + y^{'} \cos t = 7 y,$ is $\underset{―}{}$ .

3. The differential equation, $y^{″} + \sin (y) = 17 t,$ is $\underset{―}{}$ .

4. The differential equation, $\frac{d^{2} x}{d t^{2}} + e^{x} = 0,$ is $\underset{―}{}$ .

6. Which term makes the equation $y^{‴} + 3 y^{'} \sin (t) + y^{2} = 0$ nonlinear?