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Differential Equations : Definitions & Terminology

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Example Questions

Example Question #1 : Introduction To Differential Equations

State the order of the given differential equation and determine if it is linear or nonlinear.

(2-x)y'''-x^2y''+y=0

Possible Answers:

Third ordered, linear

Fourth ordered, linear

Second ordered, nonlinear

Third ordered, nonlinear

Second ordered, linear

Correct answer:

Third ordered, linear

Explanation:

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

$\\y'=\frac{dy}{dx} \\\\y''=\frac{d^2y}{dx^2} \\\\y'''=\frac{d^3y}{dx^3}$

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

(2-x)y'''-x^2y''+y=0

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

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Example Question #34 : Differential Equations

Which of the following three equations enjoy both local existence and uniqueness of solutions for any initial conditions?

1: y' = sin(y)

$2: y' = y^{2/3}$

$3: y' = y^{4/3}$

Possible Answers:

1, 2

1, 2,3

2, 3

1, 3

Correct answer:

1, 3

Explanation:

By the cauchy-peano theorem, for y' = f(t, y) , as long as f(t, y) is continuous on a closed rectangle around our starting point, we have local existence. All three functions are continuous everywhere, so they enjoy local existence at every starting point.

We can show that the solutions to differential equations are unique by showing that f(t, y) is Lipschitz continuous in y. If $\frac{\partial f}{\partial y}$ is continuous, then this will suffice to show the Lipschitz continuity.

$\frac{\partial \sin(y)}{\partial y} = \cos(y)$

$\frac{\partial y^{2/3}}{\partial y} = \frac{2}{3}y^{-1/3}$

$\frac{\partial y^{4/3}}{\partial y} = \frac{4}{3}y^{1/3}$

Note that the first and third equations are continuous for all y and t, but that the second is not continuous when y_0 = 0 . More concretely, when (y_0, t_0) = (0, 0) , both the equation y = 0 and the equation $y = \frac{t^3}{27}$ would satisfy the differential equation.

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Example Question #1 : Introduction To Differential Equations

State the order of the given differential equation and determine if it is linear or nonlinear.

(2-x)y'''-x^2y''+y=0

Possible Answers:

Third ordered, linear

Second ordered, linear

Third ordered, nonlinear

Fourth ordered, linear

Second ordered, nonlinear

Correct answer:

Third ordered, linear

Explanation:

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

$\\y'=\frac{dy}{dx} \\\\y''=\frac{d^2y}{dx^2} \\\\y'''=\frac{d^3y}{dx^3}$

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

(2-x)y'''-x^2y''+y=0

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

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Example Question #1 : Introduction To Differential Equations

Which of the following definitions describe an autonomous differential equation.

Possible Answers:

A differential equation that models growth exponentially.

A differential equation that has Eigen Values of 0.

A differential equation that does not depend explicitly on the independent variable of the equation; usually denoted or .

A differential equation that does not depend explicitly on the dependent variable of the equation; usually denoted .

Correct answer:

A differential equation that does not depend explicitly on the independent variable of the equation; usually denoted or .

Explanation:

By definition, an autonomous differential equation does not depend explicitly on the independent variable. An autonomous differential equation will take the form

f(y) y' = g(y)

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Example Question #2 : Introduction To Differential Equations

State the order of the given differential equation and determine if it is linear or nonlinear.

(2-x)y'''-x^2y''+y=0

Possible Answers:

Second ordered, nonlinear

Third ordered, linear

Second ordered, linear

Third ordered, nonlinear

Fourth ordered, linear

Correct answer:

Third ordered, linear

Explanation:

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

$\\y'=\frac{dy}{dx} \\\\y''=\frac{d^2y}{dx^2} \\\\y'''=\frac{d^3y}{dx^3}$

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

(2-x)y'''-x^2y''+y=0

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

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Example Question #2 : Introduction To Differential Equations

Find Order and Linearity of the following differential equation

$\frac{d^3y}{dt^3} + t\frac{d^2y}{dt^2} + (\cos^2{t})y = t^3$

Possible Answers:

Third Order - NonLinear

Second Order - NonLinear

Second Order - Linear

Third Order - Linear

None of the other answers.

Correct answer:

Third Order - Linear

Explanation:

This equation is third order since that is the highest order derivative present in the equation.

This is equation in linear because and derivatives appear to the first power only. t^2 and $\cos^2{t}$ do not affect the linearity.

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Differential Equations : Definitions & Terminology

Study concepts, example questions & explanations for Differential Equations

All Differential Equations Resources

Example Questions

Example Question #1 : Introduction To Differential Equations

Example Question #34 : Differential Equations

Example Question #1 : Introduction To Differential Equations

Example Question #1 : Introduction To Differential Equations

Example Question #2 : Introduction To Differential Equations

Example Question #2 : Introduction To Differential Equations

All Differential Equations Resources

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