Second-Order Control System Models One standard second-order control system model is Ко 2 3 Y(s) R(s) where п K steady-state gain the damping ratio the undamped natural ( 0) frequency , the damped natural frequency о, 1 o 125, the damped resonant frequency If the damping ratio is less than unity, the system is said to be underdamped; if is equal to unity, it is said to be critically damped, and if Ç is greater than unity, the system is said to be overdamped First-Order Linear Homogeneous Differential Equations with Constant Coefficients yay Solution, y a constant that satisfies the initial conditions 0, where a is a real constant: Ce a where C First-Order Linear Nonhomogeneous Differential Equations 1 < 0l dy x(t) = + y = Kr(t) |B t> of dt y(0) KA t is the time constant K is the gain The solution is KA (KB KA)1 - exp t or y(t) = In КА КВ t КВ

Q: . Solve: (6x + 1)y² + 3x² + 2y° %3D dr

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Q: 3.b) Evaluate the DE * 3x(xy – 2) dx + (x³ + 2y)dy = 0

A: Given that: 3x(xy-2)dx+(x3+2y)dy=0

Q: 2x²³²y" + 3xy + (2x-1)y=0

A: The given problem is to solve the given differential equation using forbenius method.

Q: 1. Solve the following first-order equation: (2.ry +3y?)dr - (2ay+a)dy = 0. %3D

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Q: Given p(x)=x -3 and &, =0.01. %3D a) Obtain the root of p(x) using False Position and Bisection…

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Q: x³Væ² + 1 dx

A: Solve the integration

Q: (2y² + 3xy – 2y + 6x)dx + x(x + 2y – 1)dy = 0

A: The differential equation is given as (2y2+3xy-2y+6x)dx+x(x+2y-1)dy=0 Let the integrating factor be…

Q: Item 7. Show that the given equation is an accurate equation and find the general solution of the…

A: Exact differential equation

Q: How can you solve this problem?  walk through step by step

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Q: Factor out the common factors. dy dx = (4x - 3)° (- 15x (2 -x³)²) + (2- x°) ³ (20(4x – 3)4) 5(4x -…

A: Topic - factorization

Q: Solve the following first order equations: (a) (x²y? + 1)dx + 2a²dy = 0, use the substitution ry =…

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Q: Obtain the roots of 25y" + 30y' + 9y = 0. m1 = m2 = %3D

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Q: Sub: Date 4y +3 dy - 4y=x`+ e* d'y dx² dx

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Q: Solve the following edo y²dx + (x² + xy + y²)dy = 0, y(0) = 1

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Q: (8x² y² -2x + 4xy² - x) dx + (2xy + x + 2y + 1) dy = 0

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Q: Find the complete solution: (cos2y - 3x²y²)dx + (cos2y − 2xsin2y — 2x³y)dy = 0

A: Given data:  cos2y-3x2y2dx+cos2y-2xsin2y-2x3ydy=0 We have to find the complete solution.

Q: -8y y² + 4xy − 2x² √ - dx

A: We can solve this using given information

Q: (D³ - 4D²)y = 5, Find Yp 5x2 4 a. O b. 5x² 4 O C. 5x² 12 O d. 5x² 12 e none

A: According to question,Given Differential equation,D3 - 4D2y = 5

Q: Solve for z. 4r 7 1 5/Ax Submit Question Iype her %3D

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Q: (√x² + y² − 2) dx + √√√x² + y² − 1) dy = 0

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Q: gression equation to predict the y ducation: hat (Y)=5x+25. What is the y

A: Given question is :     Regression equation to predict the y  :  hat (Y) = 5x+25. what is the y…

Q: dy dx - - 2x - 3y² + 2 = 0

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Q: Find the general solution of the equation: G + 6x) dx + (In x – 2) dy = 0 (x > 0) -

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Q: Solve the exact ODE (2y²x - 3) dx + (2yx² + 4)dy = 0

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Q: 2х + 3 dx Vx + 2 J 3 4 Answer: (x + 2)7 – 2(x + 2)7 + C

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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.