Ordinary Differential EquationsPlease work out the solution for x(t). 2. Please work out the solution for x(t). (Hint: Assume x(t) = ceat)(1) Find x + 4x(t) + 3x(t) = 0. (homogeneous solution)(2) Find x + 4x(t) + 3x(t) = 1. (nonhomogeneous solution)

Given: Differential equation: 1) x¨t+4x˙t+3xt=0 2) x¨t+4x˙t+3xt=1 To Find: Solution xt of the given…

Answered: Please work out the solution for x(t).…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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tion EXERCISE 8. Find the general solution of (2r-y) 2y-
Solve for the general solution * y (x + y+ 1) dx +x(x+3y+2) dy = 0 xy (x + 2y + 2) = c xy² (x + y + 1) = c Option 1 Option 2 xy (x + y+ 1) = c xy (x + 2y + 2) = c Option 3 Option 4
A 2nd order homogenous LDE has a solution xe-3x Find its general solution.
ts) Find two linearly independent solutions of 2x²y" - xy + (2x + 1)y=0, x > 0 of the form Y₁ = x¹(1+ a₁ + a₂c² + a3x³ + ...) Y₂ = x²(1 + b₁c + b₂x² + b3x³ + ...) where T1 T2. Enter T1 T2 a1 = a₂ = a3 = b₁ b₂ TE b3 N - || || || || =
Non-homogeneous Linear Equation with Constant Coefficients 4. Find the solution of (D2-4)y=4x-3e*. Ans. y=ex-x+c1e2x+C2e2x 5. Find the general solution of (D²-D)y=4e*+3ex. Ans. y=2xe*+(1/2)e2*+c1+c2e*+c3e* 6. Find the general solution of (D?+D)y=sinx. Ans. y=c1+ Cze* - (1/2)sinx - (1/2)cosx 7. Find the general solution of (D²-4D+4)y=e*. Ans. y=(c1+C2x)e2x+e*
2) Consider the predator-prey model 2 x'= ax -x²-bxy 2 y'=cy - cy² + xy where all the parameters are positive, x is the prey population, y is the predator population and they both grow logistically. Find all equilibria of the model and discuss their stability.
of the form y₁ = (1+₁+ a₂²+az³ + ...) 3₂ = x2(1+b₁x + b₂x² + b₂x³ + ...) where T₁ > T2- Enter Find two linearly independent solutions of 2x²y" - xy + (2x+1)y=0, z>0 T1 1 01 = a₂= 03 = T2= 1/2 b₁ = b₂ = b3 =
a and b please
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y" +p (t)y' +q(t)y = 0 (i), can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show (島) ( + ( +p(t) + a()y = 0 (1i). dy that dt dz dy d'y and 2 d'y d²z dy dt dz The differential equation becomes dt dr dt d'y d'z de dz dy d'y , 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H = g'(t) +2p(t)q(t) is a dy In order for equation (ii) to have constant coefficients, the coefficients of constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable,…
SOLVE Q16
Verify that if y₁ and y2 are solutions of y" + p(t)y' + q(t)y y = C₁Y₁ + C2Y2 for any constants C₁ and C₂. = 0, then so is

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Please work out the solution for x(t). (Hint: Assume x(t) = ce^t) (1) Find X + 4x(t) + 3x(t) = 0. (homogeneous solution) (2) Find X + 4x(t) + 3x(t) = 1. (nonhomogeneous solution)