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All Differential Equations Resources

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Flashcards: Definitions & Terminology

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

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

Third ordered, linear

Third ordered, nonlinear

Fourth ordered, linear

Second ordered, linear

Second ordered, nonlinear

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Flashcards: Definitions & Terminology

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

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

Your answer:

Third ordered, nonlinear

Your answer:

Fourth ordered, linear

Your answer:

Second ordered, linear

Your answer:

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.

Flashcards: Definitions & Terminology

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

1: y' = sin(y)

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

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

1, 2,3

1, 2

2, 3

1, 3

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Flashcards: Definitions & Terminology

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

1: y' = sin(y)

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

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

Your answer:

1, 2,3

Your answer:

1, 2

Your answer:

2, 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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Flashcards: Definitions & Terminology

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