Definitions & Terminology - Differential Equations

Card 0 of 6

Question

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

Answer

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,

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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Question

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

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

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

Answer

By the cauchy-peano theorem, for , as long as 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 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 . More concretely, when , both the equation and the equation $y = \frac{t^3}{27}$ would satisfy the differential equation.

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Question

Which of the following definitions describe an autonomous differential equation.

Answer

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

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Question

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

Answer

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,

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.

Compare your answer with the correct one above

Question

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

Answer

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,

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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Question

Find Order and Linearity of the following differential equation

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

Answer

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. and $\cos^2{t}$ do not affect the linearity.

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