Differential Equations Exam Questions and Answers

Differential Equations Exam Questions and Answers What is a differential equation? - Answer-An equation which relates several quantities and their (1st, 2nd, 3rd, etc) derivatives y' = 0. What is the rate of change of y? What are the solution(s)? - Answer-Nothing. Y is a constant quantity. The solutions are y(x) = C, where C is a constant. y' = x. Solution(s)? - Answer-dy/dx = x, so y(x) = 1/2(xˆ2) + C y = 2x + [cos(xˆ2)]. Solution(s)? - Answer-sinxˆ2 + C dy/dx = y. Solution(s)? - Answer-y(x) = Aeˆx (because y(x) = eˆx doesn't satisfy y'= y). A is a constant. A 1st order equation (meaning only x', not x') always has the form... For example, x'= tˆ2(x) has the form f(t,x) - Answer-x = f(t,x), f(t,x) = tˆ2(x) What does f(t,x) tell us? - Answer-The slope of the line at a point t,x What are the 3 types of diff equations? - Answer-1) Calc I style x'=(t only) 2) Autonomous x=(x only) 3) Separate equations x'= (x stuff only) + (t stuff only) How to solve a calc 1 style equation like x = tˆ2 + 2t + 1?? - Answer-Take anti derivative x = f(tˆ2 + 2t + 1dt), x(t) = tˆ3 /3 + tˆ2 + t + C How to solve an autonomous equation? x'= x - Answer-dx/dt = x. Shuffle x to one side, dt to the other. dx/x = dt. Take the integral. ∫dx/x = ∫dt -->log(x) = t+C. Solve for x. eˆ(logx) = eˆ(t +C). X = eˆ(t+C) = eˆt * eˆc = Aeˆt x'= √x. Solution? - Answer-x = (tˆ2)/4 + Ct + Cˆ2. HOW? dx/dt = sqrt(x) -- dx/(sqrt(x)) = ∫dt -- ∫xˆ(-1/2)dx = t + C -- 2xˆ(1/2) = t+ C -- xˆ(1/2) = (t/2) + (C/2) -- x = (t/2 + C)ˆ2 = tˆ2 /4 + Ct + Cˆ2