Chapter 4 Problem Notes Updated 2015 4.1-6: ========================================================================= Make the position-velocity substitution u1=x(t), u2=x'(t), u3=y(t), u4=y'(t) This changes the coupled system of two second order DE into a first order system u' = Au for some 4x4 matrix A. The book's answer is u'=Au in scalar form using x1,x2,x3,x4 instead of u1,u2,u3,u4. 4.1-8: Make the position-velocity substitution u1=x(t), u2=x'(t), u3=y(t), u4=y'(t) This changes the coupled system of two second order DE into a first order system u' = Au for some 4x4 matrix A. The book's answer is u'=Au in scalar form using x1,x2,x3,x4 instead of u1,u2,u3,u4. Details: First, u1' = u2. To find an equation for u2', proceed as follows: x'' = u1'' = u2' and x'' = -3x' - 4x + 2y = -3u2 - 4u1 + 2u3. Then u2' = -4u1 - 3u2 + 2u3 + 0u4. The other quations are found similarly. The matrix A is 4x4, having the form 0 1 0 0 -4 -3 2 0 * * * * * * * * 4.1-18: Skip the direction field work. Submit only the details for the general solution and the evaluation of initial conditions. Start by writing the system as u'=Au. [x(t)] [0 -1] u = [ ], A = [ ] [y(t)] [10 -7] The characteristic equation of A is r^2 + 7r + 10 = 0 with roots -2,-5. The theory says that u1 and u2 are solutions of the second order equation obtained from this characteristic equation [namely w''+7w'+10w=0] and therefore u1=x(t) is a linear combination of the atoms exp(-2t) and exp(-5t). This gives the answer x(t) = c1 exp(-2t) + c2 exp(-5t) y(t) = -x'(t) from the DE system = 2c1 exp(-2t) + 5c2 exp(-5t) 4.1-20: Skip the direction field work. Submit just the details for the general solution and the evaluation of initial conditions. Start by writing the system as u'=Au. [x(t)] [ 0 1] u = [ ], A = [ ] [y(t)] [-9 6] The characteristic equation of A is r^2 -6r + 9 = 0 with roots r=3,3. The theory says that u1 and u2 are solutions of the second order equation obtained from this characteristic equation [namely w''-6w'+9w=0] and therefore u1=x(t) is a linear combination of the atoms exp(3t) and t exp(3t). This gives the answer x(t) = c1 exp(3t) + c2 t exp(3t) y(t) = x'(t) from the DE system = 3 c1 exp(3t) + 3 c2 t exp(3t) + c2 exp(3t) =========================================================================