Consider the differential equation d dx(t) = ax+y+x³ d =x-y, First, sketch the nullclines of the differential equation, determine any equilibria, and linearise the system near these equilibria as a function of the parameter a. Then, determine the local stability of the origin as a function of the model parameter a.

Q: A force of 640 newtons stretches a spring 4 meters. A mass of 40 kilograms is attached to the end of…

A: Given that Force (F) = 640 Newtons Distance (x) =4 meters Mass (m) = 40 kilograms Upward velocity…

Q: dy/dt= y(y-0.5)(y-3)(sin(t2)-2) Determine the equilibria.

A: In this question, we Determine the equilibria. To explain why the Existence and Uniqueness theorem…

Q: The relative displacement u(t) of a single-storey shear building subjected to an earthquake ground…

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Q: Use the "mixed partials" check to see if the following differential equation is exact. If it is…

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Q: is poured into the tank a te of 2 liters/min. Let y(t ial model for (t) and gol

A: The figure as,

Q: Draw a direction field for the differential equation y' = -y(2-y). Based on the direction field,…

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Q: P(t)y'' + Q(t)y' + R(t)y = S(t) is this a linear equation?

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Q: Use the elimination method to find a general solution for the given linear system, where…

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Q: y=3 is an equilibrium solution for the d.e. y'-y=3 Select one: True O False

A: We have to check

Q: 1. Provide at least 6 vectors of the slope field of the differential equation y and use them to…

A: The given differential equation is y'=x-y. We need to find the solution curve through the point 1,2…

Q: x' Find the solution to the system of differential equations: y' x(0) = 5 y(0) = 2 = y - x + t, Y,…

A: To find the solutions of the below differential equation,  a) x' = y - x + t         ,     x(0) = 5…

Q: 1. Describe each of the following equations, giving its order and telling whether it is ordinary or…

A: 1)Given DE is y"+3(y')2+4y=0. We find it's order and decide whether it is ordinary or partial and…

Q: Using the autonomous equation dy/dx=f(y), how it is possible to obtain information about the…

A: Image is attached with detailed solution.

Q: Given the differential equation y' = x (x - 2) (2+ x)² a) State the equilibrium solutions. b) Draw…

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Q: Find the linearly independent solution of the equation y"+xy'+y=0 around the point x=1.

A: We will find out the required solution.

Q: Find all equilibria of the continuous-time dynamical system given by i = cos x, x e R. Sketch the…

A: Given: Let's find all of the equilibria of the continuous-time dynamical system x˙=cosx, x∈ℝ.Draw…

Q: t?y" +2t y' – 3y = tant , as a system of linear differential equation is: Select one: a. tant b.…

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Q: d²x dr? dx + 2 - 3x + x = 0 dt (a) Convert this second-differential equation into a first-order…

A: We are given : dx2dt2+2dxdt-3x+x3=0---(1) v=dxdt---(2)⇒dvdt=dx2dt2--(3) Using (2),(3) in (1), we…

Q: Show that a linear equation dy a(x)- + b(x)y = c(x) d.x %3D is exact if a' (x) = b(x).

A: Given linear differential equation axdydx+bxy=cx.....1 We have to show that the given differential…

Q: Consider the scalar differential equation d Define the potential V(x) by Use the chain rule to show…

A: Note: We will be solving part (H2) only as asked.Given that .Define potential function  defined as…

Q: The function u depends on two independent variables: u= u(x, t). Find the general solution of the…

A: Given: The function u depends on two independent variables: u=u(x,t). PDE: ∂2u∂t2+∂u∂t+u=2t-x. To…

Q: Write the given third order linear equation as an equivalent system of first order equations with…

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Q: determine the dependent variables and independent variables Identify the type and determine the…

A: The given differential equation is - 5+(y''')24=3z3y2+(y'')343

Q: Consider the Riccatti equation  v'= v^2-7v/x +8/x^2 a) Use the substitution  v= y'/y  obtain the…

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Q: What is the equilibrium solution of the equation y '(t) = 2y - 4? Is itstable or unstable?

A: The equation given is y't=2y-4. To find the equilibrium solution of the equation y't=2y-4.…

Q: dx Graph the slope field for the equation dt -r(x² - 4) and attach it to the solution. Draw (by hand…

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Q: Solve for the 2nd order nonhomogeneous linear equation: y"+12y'+37y=10e-4t     ; y(0)=4,  y'(0)=0

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Q: A force of 660 newtons stretches a spring 3 meters. A mass of 55 kilograms is attached to the end of…

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Q: Use the elimination method to find a general solution for the given linear system, where…

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Q: Solve 2y-y² sec²(xy²) + (2x−2xysec² (xy²)) y'= 0; y(1) = 2

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Q: x² = −wy + xy² + x²y+y³ y' = wx+y³x³ = xy²

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Q: Consider the general first-order linear equation y'(t) + a(t)y(t) = f(t). This equation can be…

A: Given below the detailed solution