Let A and B be III x II matrices. Show th;!t A is row equivalent to B if and only if AT is column equivalent to

Math246 Lecture 6: Second Order Differential Equations a y +b y cy=h(t) Now, we will be looking at second order differential equations of the form . Of particular interest is when h(t)=0. Let’s look at some examples. General Problems y +25y=0 1. What function essentially doesn’t change when you take its derivative Yes, the x exponential function e . So now all we need to do is get the exponent correctly. One e5t 25e 3t solution is . If you take the derivative twice, you end up with . Another 5t −5t solution is e−5t . A more general solution is y (t=ae +be . 2. Let’s now solve the same problem with some initial conditions. y(0)=5 and y'(0)=20 . So we need to take the derivative of the general solution first. ' 5t −5t y t =5ae −5be Now we plug in the initial conditions. 20=5a−5b 5=a+b a=5−b b So and we can plug that into the first equation and solve for . 20=25−5b−5b=25−10 b 1