2. Consider the nonlinear system of differential equations dx = r*y – xy dt dy = y – e" dt (a) Determine all critical points of the system. (b) For each critical point not on the y-axis: i. Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system. ii. Find the general solution of the linearised system using eigenvalues and eigenvectors.

Q: How to find the characteristic equation of the system: dx(t)/dt=Ax(t) + Bu(t)

A: To find the characteristic equation of the system:

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Q: Solve the following system of differential equations: dx dy dt dt dx dy dy_x-y=1 dt - 2-2x+4y=t, dt

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Q: The input-output relationship of a system (mechanical or electrical) is described by the following…

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A: Given, Y'=AYYt=xtytY't=x'ty'tA=1-441

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Q: 1. Write down the system of differential equations for the gradient flow associated to V (x, y) : xy…

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Q: -2 -3 Y dY dt 3 -2 (a) Show that the two functions Y1(t) = e=#(cos 3t, sin 3t), Y2(t) = e=#(- sin…

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Q: 6. Find the general solution to the system of equations dx dy + dt dt 2t + +x+y=0 d²x dx dt² dt

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Q: D²x – Dy = t x(1) = 1, y(1) = 0, D = 4. %3D (D + 3)x + (D + 3)y = 3 dt

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Q: Consider the model of two competing populations defined by (dx = x(2 – 0.4x – 0.3y) dt |dy - = y(1 –…

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Q: 5. Consider the new linear system of DES. where x(0) = 4, and y(0) = 1. dx dt dy dt = = 2x+4y -9x -…

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Q: Consider the system {{ = - sin(x) (a) Find all the critical points (equilibrium solutions). (b) Find…

A: (a) Given system of equations are dxdt=y and dydt=-sinxthat is X'=AX where A=01-sinx0Critical points…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.