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Questions and Answers
An autonomous equation explicitly contains the independent variable, T.
False
The form of an autonomous equation can be represented as dY/dT = F(Y), where Y is a function of T.
True
The equilibrium points of autonomous equations are not significant in determining stability.
False
To check the stability of an equilibrium point, we can use the stability test involving whether F(Y*) is positive at the equilibrium point Y*.
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If F(Y*) is negative, the equilibrium point is unstable.
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Converting an autonomous equation into Dy = F(Y)dY allows for easy integration to find the solution.
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If F(Y*) = 0, the equilibrium point is stable.
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One of the theorems discussed ensures the existence and uniqueness of solutions for autonomous differential equations if F(Y) is continuously differentiable and positive except for the endpoints.
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The initial condition Y0 in the solution of an autonomous equation refers to the initial value of the independent variable, T.
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Autonomous equations cannot be solved using separable variables.
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Study Notes
- The text discusses autonomous equations in the context of differential equations and separable equations.
- An autonomous equation is a type of differential equation where the equation does not explicitly appear, and it can be solved using separable variables.
- The form of an autonomous equation is Dy/DT = F(Y), where Y is a function of T.
- To find the solution of an autonomous equation, we convert it into Dy = F(Y)dY, then integrate both sides to get Y - Y0 = ∫F(Y)dY, where Y0 is the initial condition.
- The equilibrium points of autonomous equations are important as they can be stable or unstable.
- To check the stability of an equilibrium point, we can use the stability test, which involves checking whether F(Y*) is positive or negative at the equilibrium point Y*.
- If F(Y*) is positive, the equilibrium point is unstable. If F(Y*) is negative, the equilibrium point is stable.
- If F(Y*) = 0, the equilibrium point is neutral.
- The text also discusses two theorems that ensure the existence and uniqueness of solutions for autonomous differential equations.
- The first theorem states that if F(Y) is continuously differentiable, positive except for the endpoints, and the initial value is chosen within the interval of definition, then the autonomous differential equation has a solution.
- The second theorem states that if F(Y) is continuous, has only one root in the open interval, and the integral of F(Y) diverges, then the autonomous differential equation has a unique constant solution.
- The text provides examples of solving autonomous differential equations by separating variables.
- In the first example, the text models the production and natural loss of red blood cells using an autonomous differential equation.
- In the second example, the text calculates the amount of money accumulated from an initial investment over a long period with continuous compounding interest.
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Description
Explore the concept of autonomous equations in the context of differential equations and separable variables. Learn about finding solutions, stability tests, equilibrium points, and the existence theorems for autonomous differential equations. Find examples of solving autonomous differential equations by separating variables.
