Find the steady-state solution, w(x), of each of the following initial-value problems. Also findthe initial-value problem satisfied by the transient solution, v(x, t) = u(x,t) – w(x). Do notsolve the problem for the transient solution.Ut =kuxx — В(и — иm), 0 0--и(0, t) — ит и(L,t) 3D 0, t > 0u(x,0) = f (x), 0 < x < L3.

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Answered: Find the steady-state solution, w(x),…

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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A second-order chemical reaction involves the interaction of onc molccule of a substance P with one molecule of another substance Q to produce one molecule of another substance X. Suppose that p and q are the initial concentrations of P and Q and let r(t) be the concentration of X at time t. Then p- I(t) and q -r(t) are the concentrations of P and Q at time t, and the rate at which the reaction occurs is given by the equation dr *(I - b)(I – d)o= dt where a is a positive constant. Solve the initial value problem with r(0) = 0 assuming p # q. Answ: r(t) = Pq(e®(q-p)t _1 qea(q-p)t-p , you may nced to manipulate your answer to get this.
Make a program in matlab that graphs the solution of the partial differential equation with u(0,t) = 0, u(π,t) = 1, u(x,0) = 0. The program should graph the solution u(x,t). The program must request the evolution time.
The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P, has doubled in 5 years. Suppose it is known that the population is 12,000 after 3 years. What was the initial population Po? (Round your answer to one decimal place.) Po What will be the population in 10 years? (Round your answer to the nearest person.) persons How fast is the population growing at t= 10? (Round your answer to the nearest person.) persons/year Need Help? Read It Watch It
(c) Show that if f is differentiable and W is a function, then every solution g to fd - S'g = w is of the form W (t) dt. (1) Hence, given a single solution f, you can find all solutions g to the original equation y" + a(t)y' + b(t)y = 0 using (1).
2. Consider the Bernoulli equation = -e *y, y(0) = 1. 6. The substitution v = yl-" which transforms the Bernoulli equation into the first order linear equation for v. (a) Find n and v. (b) Find the linear equation for v. (c) Find the general solution for v. (d) Using the initial condition y(0) = 1 to determine the coefficient in v. (e) Find the unique solution y(x) to the initial value problem.
Find the values of c¡ and c2 so that the given functions will satisfy the prescribed initial conditions y(x) = c1x+ c2 + x² – 1; y(1) = 1, y'(1) = 2
4. (a) Consider the ODE y(1+r³)y = r². Determine where in the xy-plane existence and uniqueness issues to an associated initial value problem may occur. (b) Solve the IVP y(1+x³)y′ = x², y(0) = yo and determine how the interval in which the solution exists depends on the initial value yo.
You can verify that the differential equation: 2t°y" - 4t(t + 1)y + 4(t +1)y= 0, t> 0 has solutions y1 = 2t and y2 = 3t exp (2t). a. Compute the Wronskian W between y1 and y2. W (t) = | 12-e2t b. The solutions y1 and y2 form a fundamental set of solutions because there is a point to where W(to) + 0: W # 0. Note: It simply wants you to find some number to to put into the first answer blank and plug into the Wronskian to get something nonzero, which goes into the second answer blank.
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(a) Draw a diagram that depicts the flow process described above. Let QX(t) and QY(t), respectively, be the amount of salt in each tank at time t. Write down differential equations and initial conditions for QX and QY that model the flow process. Justify if the system of differential equation is homogeneous or non-hompgeneous
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