Question 4 dr Consider the ODE = ex ox³, where e and o are constants. This ODE is important in the study of fluid flow.

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Question 4 Dr Consider The Ode Ex Ox Where E And O Are Constants This Ode Is Important In The Study Of Fluid Flow 1 (171.16 KiB) Viewed 10 times

Question 4 dr Consider the ODE = ex ox³, where e and o are constants. This ODE is important in the study of fluid flow. It dt is not linear. However, it can be made linear with the substitution y(t) = [x(t)]-2. (Throughout this problem, you can assume that t> 0 and r(t) >0 so that we are never dividing by zero.) 1. Solve the formula y(t) = [x(t)]-2 for x. Take the derivative of both sides with respect to t to find a formula for de in terms of and y. (You only need to use the formula y = x-2 for this part, not the differential equation. This should be a very easy problem.) 2. Substitute the formulas you just found for de and a into the ODE. You should end up with a new differential equation for y(t). This new differential equation should be linear. 3. Find the general solution for y. (That is, solve the linear ODE that you found in part 2.) 4. Use your general solution for y to find the general solution for r. (This should be very easy.) 5. In general, a differential equation of the form a(t) dr + b(t)x= c(t)r" where a(t) #0 for any t and n # 0,1 is called a Bernoulli equation. Show that the substitution y(t) = [x(t)]¹-n turns a Bernoulli equation into a linear ODE. Question 5 Consider the differential equation da dt = f(t, x), where f and are both continuous functions for all t and r. Define (t) as the solution to this differential equation with initial condition r(to) = a and define (t) as the solution to this differential equation with initial condition z(to) = b, where to, a and b are constants. Prove that if a b then the graphs of xa and x, never cross. That is, prove that there is no time t* such that xa (t*)=xt(t*).