Question 2(a) Determine whether the set of functionsf₁(x) = x, f₂(x) = x − 1, f3(x) = x+3is linearly independent on the interval (-∞0,00).(b) Consider the 2nd order linear differential equationxy" + y = 0.(i) Given that y₁ = Inz solves the equation, use reduction of order to obtain theother solution 2.(ii) Show that the set of solutions {1, 2} is a fundamental set ofsolutions.(iii) Hence, or otherwise, write down the general solution of the equation.,

Answered: Image /qna-images/answer/a0dfa835-b4ac-4610-8a93-39c9d5cbddaa.jpg

Answered: (a) Determine whether the set of…

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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Verify that the function c,(x + y) = xe is a solution of the differential equation (x2 + y dx + (x +(x²- xy kdy = 0 ху 0.
Let the solution of the differential equation (1 + 2y2) dx + xydy = 0 be f(x,y) = c, where f(x,y) is an algebraic relation in x and y. Determine the value f C if x=15 and y=15.
Verify that the indicated function y = p(x) is an explicit solution of the given first-order differential equation. (y - x)y' = y - x + 2; y = x + 2√x + 5 When y = x + 2√√x + 5, y' Thus, in terms of x, (y-x)y' = y-x+ 2 = Since the left and right hand sides of the differential equation are equal when x + 2√x + 5 is substituted for y, y = x + 2√x + 5 is a solution. Proceed as in Example 6, by considering simply as a function and give its domain. (Enter your answer using interval notation.) Then by considering as a solution of the differential equation, give at least one interval I of definition. O (-00,-5) O (-10, -5] O (-10, 5) O [-5, 5] O (-5,00)
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Now consider a wildlife population that is exposed to an infectious disease, with densities of susceptible and infected individuals given by S and I respectively. Use the information below to write down a set of ordinary differential equations that describe this system. • Susceptible hosts can reproduce at rate b and all offspring are born susceptible, but the disease prevents infected hosts from reproducing. • All hosts die at natural mortality rate d and infected hosts suffer further mortality at rate α. • Infectionoccursthroughdirectcontactofsusceptibleandinfectedhosts,withtheforceofinfectiongiven by βI. • Infected hosts can recover from disease at rate γ, but they do not gain immunity and instead return to being susceptible.
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