Answer Happy

1-1. (15 points) Given the following first order ODE (ey + 2x)dx + eydy = 0 Eq. (Q1-1) (a). (5 points) A student gives a general solution xey + 2x = c, Sol. (Q1-1) Where c is an arbitrary constant. Verify whether Sol. (Q1-1) is the solution of Eq. (Q1-1) Justify your answer. (b). (10 points) Solve y(x) for the initial condition y(0) = 1. 1-2. (10 points) (a). (7 points) Solve the following Bernoulli equation y'(x) – y(x) = y4(x) Eq. (Q1-2) (b). (3 points) Based on the solution found in 1-2(a), find the limit of y(x) when x → 0. Question 2 (25 points) 2-1. (12 points) Given the following second order ODE x?y" – xy' + y = 0 Eq. (Q2-1) (a). (5 points) By inspection, it is known that y = x is a solution of Eq. (Q2-1). Find another solution of Eq. (Q2-1) that is linearly independent with y = x; (b). (2 points) Use the Wronskian of the solution y = x and the other solution you have obtained above to prove that the two solutions are linearly independent; and (c). (5 points) Solve the initial value problem with Eq. (Q2-1) and y(1) = 3 and y'(1) = 2. 2-2. (13 points) Solve the following initial value problem y'' – 2y' + y = 2e* + 2sinx and y(0) = 0, y'(0) = 0. Eq. (Q2-2)