a. Draw a direction field and sketch a few trajectories. b.Express the general solution of the given system of equations in terms of real-valued functions. c.Describe the behavior of the solutions as t → ∞.4.x′=(12−5−1)x

Q: Obtain the homogeneous linear equation satisfied by the following given function 1. y =x2 (e5x + 4)

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

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Q: A. Determine the general solution of the following nonhomogeneous linear equations using the method…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: 3. Given that f(x) = x is a solution of (1 – x² )y" – 2xy + 2y = 0 within -1<x<1 find a second…

A: Given: (1-x2)y"-2xy'+2y=0  within -1&lt;x&lt;1:

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Q: Apply Euler-Cauchy Runge-Kutta method for the equation: and obtain y' = 29 t y (1) =1 y, y, and y. 3…

A: Hint: Formula for finding n-th approximate solution of IVP  dydx=f(x,y), y(x0)=y0  is…

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Q: , solve this D using the differential operator (D - 13D2 + 36)y = 2e2x + 3e-2x + 4e3x + 5e-3x + 10e*

A: Using method of undetermined coefficients

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Q: 7. Show that y(t) = c,e¬# +czte¬2# is a solution to y" +4y/ +4y = 0. Determine the constants c and…

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Q: Consider the following IVP: y"-3y'+2y = e3t; y(0) = 0, y'(0) = 0 Find the general solution. A y(t) e…

A: Given differential equation is y'' - 3y' + 2y = e3t ; y(0) = 0, y'(0) = 0

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Q: If these two y's, y1=e^tcos(t) and y2(t)=e^tsin(t) are both solutions to y''+j(t)y'+k(t)y=0, how do…

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Q: Q5) Solve the system dx +y = et dt dy dt

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Q: Let a ∈ R and find the solution to y'(t) - a y(t) = 0.

A: The given differential equation is dydt-ay=0 , a∈R.we will slove this by the method of  separartion…

Q: show that the solution y(t)=e^2tsin(t) is linearly independant.

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Q: 23. Find the particular solution that satisfies the initial condition. y'+ 3x²y = 3x². Initial…

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Q: 7x The orthogonal trajectories of y = ce'",where c is arbitrary constant. is 2x + 2C = 7y? O -2x +…

A: Given,  y = c e^(7x)    y e^(-7x) = c then get the derivative   dy/dx e^-7x + y {-7 e^(-7x)} = 0…

Q: x²y" +xy' – 4y = 2x*

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Q: 3. Determine the general solution of the following nonhomogeneous linear equations using the method…

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Q: Obtain the homogeneous linear equation satisfied by the following given function 1. y = 4e2x + 3e¬x…

A: The homogeneous linear equation satisfied by the given equation: Part(1):   y=4e2x+3e−x⇒3e−x=y−4e2x…

Q: a²u a2u 1. əx² au + = 0 az? ду? 2. dt = w(x) dx4 3.x d²y y dt2 d?x = C1 - dt2

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Q: Solve the IVP y + 2y = 0 if t≥ 1 y(0) 1. Is the solution continuous? = "

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Q: Obtain the general solution of the following: A. (D2 − 1)y = ex + 1 B. (D2 + 1)y = sec3x

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Q: Solve the following equation dy - 4t+ 6y = t3 +1 dt2 dt

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Q: Determine the orthogonal trajectories for x2 = ky³. A) 2x2 + 3y2 = c C) 2x2 – 3y? = C - D1.2

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Q: first finding particular intergral and complimentary solution.

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Q: Q5) Solve the homogenous system dy = 2x + 3y dt dx 2х + у dt

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Q: Use the method of undetermined coefficients to find a general solution to the system x'(t) = Ax(t) +…

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Q: The equation below is said to be A. Linear in x O B. Linear in y C. Linear in both x and y O D.…

A: We will find out the required value.

Q: 22.Consider the equation y″ − y′ − 2y = 0. a.Show that y1(t) = e−t and y2(t) = e2t form a…

A: Given the equation y''-y'-2y=0. We need to show that y1(t) = e−t  and  y2(t) = e2t form a…

Q: - Find the solution of the separable DE 2ydy = (x + 1)dx. a. y- +x x+c c y=x'+x+C d. y=x + 3x + C b.…

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Q: 1. Find the order and degree of each DE, and state whether it is linear or nonlinear and why? a.…

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Q: Find the second linearly solution of the D.E (2x +1) y" – 4(x+1) y' + 4y = 0, given that : y,(x) =…

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Q: Consider the following system of equations in R²→R? y + 2xỉ + x3 - yıY2 = 15 2y + xỉ + x + YıY2 = 38…

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Q: Solve the non-homo . D.E. 1.y" + у %3D Зе2х

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Q: 2. Determine whether or not y = Ce2t + Cze-4t + (4/7)e3t is a solution to the differential equatior…

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Q: (a) If b – 4ac = 0 and y1 = em solves ay" + by' + cy = 0, use reduction of order to get a linearly…

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Q: 21/ solve one of the following Linear system: dx 1. - y = t? dt 2+ 4x = 1 dt

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Q: 17. Find the solution of the separable DE 2ydy = (x² + 1)dx. a. y = E+x+C +x +C 3 b. y =+x+C c. y =…

A: As we know that; The Power rule; ∫xndx=xn+1n+1+C ; n≠-1 From the given differential equation;…

Q: Suppose a population is modeled by the equation P+1 = aP;e¯rP. where a and r are positive real…

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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.