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Chapter 5: Q9E (page 259)

In Section 3.6, we discussed the improved Euler鈥檚 method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

Short Answer

Expert verified

The result is:

xi,n+1=xi,n+h2fi(tn,x1,n,x2,n........,xm,n)+fi(tn+h,x1,n+hf1(tn,x1,n,x2,n........,xm,n)........,xm,n+hfm(tn,x1,n,x2,n........,xm,n)

Step by step solution

01

Use Euler’s method

Here given Euler鈥檚 method of the differential equation:

So,yn+1=yn+h2f(xn,yn)+f(xn+h,yn+hf(xn,yn))

苍=0,1,2,鈥�..补苍诲 xn+1=xn+h.

Now,

x'1(t)=f1(t,x1,x2........,xm)x'2(t)=f2(t,x1,x2........,xm)..x'm(t)=fm(t,x1,x2........,xm)

02

Solve for every i

For every I from 1 to m, then;

tn+1=tn+hxi,n+1=xi,n+h2fi(tn,x1,n,x2,n........,xm,n)+fi(tn+h,x1,n+hf1(tn,x1,n,x2,n........,xm,n)........,xm,n+hfm(tn,x1,n,x2,n........,xm,n)

This is the required result.

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Most popular questions from this chapter

A house, for cooling purposes, consists of two zones: the attic area zone A and the living area zone B (see Figure 5.4). The living area is cooled by a 2 鈥� ton air conditioning unit that removes 24,000 Btu/hr. The heat capacity of zone B is12F鈭�per thousand Btu. The time constant for heat transfer between zone A and the outside is 2 hr, between zone B and the outside is 4 hr, and between the two zones is 4 hr. If the outside temperature stays at 100掳F, how warm does it eventually get in the attic zone A? (Heating and cooling buildings was treated in Section 3.3 on page 102.)

Sticky Friction. An alternative for the damping friction model F = -by鈥� discussed in Section 4.1 is the 鈥渟ticky friction鈥� model. For a mass sliding on a surface as depicted in Figure 5.18, the contact friction is more complicated than simply -by鈥�. We observe, for example, that even if the mass is displaced slightly off the equilibrium location y = 0, it may nonetheless remain stationary due to the fact that the spring force -ky is insufficient to break the static friction鈥檚 grip. If the maximum force that the friction can exert is denoted by m, then a feasible model is given by

\({{\bf{F}}_{{\bf{friction}}}}{\bf{ = }}\left\{ \begin{array}{l}{\bf{ky,if}}\left| {{\bf{ky}}} \right|{\bf{ < }}\mu {\bf{andy' = 0}}\\\mu {\bf{sign(y),if}}\left| {{\bf{ky}}} \right| \ge {\bf{0andy' = 0}}\\ - \mu {\bf{sign(y'),ify'}} \ne 0.\end{array} \right.\)

(The function sign (s) is +1 when s 7 0, -1 when s 6 0, and 0 when s = 0.) The motion is governed by the equation (16) \({\bf{m}}\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - ky + }}{{\bf{F}}_{{\bf{friction}}}}\)Thus, if the mass is at rest, friction balances the spring force if \(\left| {\bf{y}} \right|{\bf{ < }}\frac{\mu }{{\bf{k}}}\)but simply opposes it with intensity\(\mu \)if\(\left| {\bf{y}} \right| \ge \frac{\mu }{{\bf{k}}}\). If the mass is moving, friction opposes the velocity with the same intensity\(\mu \).

  1. Taking m =\(\mu \) = k = 1, convert (16) into the firstorder system y鈥� = v (17)\({\bf{v' = }}\left\{ \begin{array}{l}{\bf{0,if}}\left| {\bf{y}} \right|{\bf{ < 1andv = 0}}{\bf{.}}\\{\bf{ - y + sign(y),if}}\left| {\bf{y}} \right| \ge {\bf{1andv = 0}}\\{\bf{ - y - sign(v),ifv}} \ne 0\end{array} \right.\) ,
  2. Form the phase plane equation for (17) when v 鈮� 0 and solve it to derive the solutions\({{\bf{v}}^{\bf{2}}}{\bf{ + (y \pm 1}}{{\bf{)}}^{\bf{2}}}{\bf{ = c}}\).where the plus sign prevails for v>0 and the minus sign for v<0.
  3. Identify the trajectories in the phase plane as two families of concentric semicircles. What is the centre of the semicircles in the upper half-plane? The lower half-plane?
  4. What are the critical points for (17)?
  5. Sketch the trajectory in the phase plane of the mass released from rest at y = 7.5. At what value for y does the mass come to rest?

In Problems 25 鈥� 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), z(t).

x'=3x+y-z,y'=x+2y-z,z'=3x+3y-z

In Problems 3鈥�6, find the critical point set for the given system.

dxdt=y-1dydt=x+y+5

In Problems 3鈥�6, find the critical point set for the given system.

dxdt=x-y,dydt=x2+y2-1
See all solutions

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