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(2) Why First-Order Systems (of a Specific Form) Are Sufficient: In class I stated that all systems of differential equations can be turned into first-order systems. And I wrote that first-order systems can be written in the form: For 3-by-3 system, the form is: r-f(r, y,z,t) y- g(x, y, z, t) , = h(z,y,z, This can be generalized to any number of unknown functions. a) Notice that r', y', and z are not included in the functions f, g, and h in the general form given in Lebl's text. They are not, as is usual, because for the equations that we will consider, if there were derivatives of these functions on the right, the equations could be algebraically re-arranged to have no derivatives on the right side. For instance, use algebra to rearrange the following system into the form abpve; 2'-y+3 +e First, you'll want to use substitution to replace r in the equation for z'. Then, use the new equation for z' to substitue into the equation for y'. Then solve for y'. Then use substitution again (twice) to get a system of the desired form (Eq. 2) (b) The more important way that systems of the form given at the start of this problem are sufficient for all systems is that higher-order systems can be reduced to first-order systems with more unknown functions. To illustrate this, you a first-order system for four unknown functions. The equation for the bending of a beam under a certain load is ı will turn one fourth-order equation into El-= s, a certain load is 3 El-=s, where w(s) is the displacement from straight (or deflection) of the beam s units from the end of the beam. E and I are constants related to the stiffness of the beam. Turn this into a first-order system using the following steps Define a function z(s) as the derivative of w(s) . Define a function y(s) as the second derivative of w(s) . Define a function z(s) as the third derivative of w(s) Give in terms of w, x, y, and z . Give r in terms of w, x, y, and z Give y in terms of w, z, y, and z . Give z' in terms of w, z, y, and z, using the differential equation Eq. (3) You should now have 4 very simple equations in the form given in Eq. 2. (This problem is quite simple once you understand the idea. See recitation notes or the bottom of pg. 104 in Lebl

Answer 1

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