Math  /  Calculus

Question1,2,3,4,51,2,3,4,5, and 6 Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous.
1. dx/dt=xy,dy/dt=3ty+xd x / d t=x-y, \quad d y / d t=-3 t y+x

HIDE ANSWER Answer: Linear, homogeneous, nonautonomous
2. dy/dx=2y,dz/dx=xz+3d y / d x=2 y, d z / d x=x-z+3
3. dy/dt=3yz2z,dz/dt=2z+5yd y / d t=3 y z-2 z, \quad d z / d t=2 z+5 y

Studdy Solution

STEP 1

1. A system of differential equations is autonomous if the independent variable (often time t t ) does not explicitly appear in the equations.
2. A system is linear if each equation is a linear combination of the variables and their derivatives.
3. A linear system is homogeneous if there are no constant or non-zero terms independent of the variables.

STEP 2

1. Analyze the system of equations to determine if it is autonomous or nonautonomous.
2. Analyze the system of equations to determine if it is linear or nonlinear.
3. If the system is linear, determine if it is homogeneous or nonhomogeneous.

STEP 3

Analyze the system dxdt=xy,dydt=3ty+x \frac{dx}{dt} = x - y, \quad \frac{dy}{dt} = -3ty + x .
- Check for the presence of the independent variable t t in the equations. - The second equation contains t t , so the system is nonautonomous.

STEP 4

Analyze the system to determine if it is linear.
- Both equations are linear combinations of the variables x x and y y . - There are no products or nonlinear functions of x x and y y .
The system is linear.

STEP 5

Determine if the linear system is homogeneous.
- There are no constant terms or terms independent of x x and y y .
The system is homogeneous.
For the first system, the classification is: Linear, homogeneous, nonautonomous.

STEP 6

Analyze the system dydx=2y,dzdx=xz+3 \frac{dy}{dx} = 2y, \quad \frac{dz}{dx} = x - z + 3 .
- Check for the presence of the independent variable x x in the equations. - The second equation contains x x , so the system is nonautonomous.

STEP 7

Analyze the system to determine if it is linear.
- Both equations are linear combinations of the variables y y and z z . - There are no products or nonlinear functions of y y and z z .
The system is linear.

STEP 8

Determine if the linear system is homogeneous.
- The second equation has a constant term +3 +3 .
The system is nonhomogeneous.
For the second system, the classification is: Linear, nonhomogeneous, nonautonomous.

STEP 9

Analyze the system dydt=3yz2z,dzdt=2z+5y \frac{dy}{dt} = 3yz - 2z, \quad \frac{dz}{dt} = 2z + 5y .
- Check for the presence of the independent variable t t in the equations. - Neither equation contains t t , so the system is autonomous.

STEP 10

Analyze the system to determine if it is linear.
- The first equation contains a product of variables 3yz 3yz , which is nonlinear.
The system is nonlinear.
For the third system, the classification is: Nonlinear, autonomous.

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