DIRECTIONS: Complete all problems on notebook paper, scan/t…
DIRECTIONS: Complete all problems on notebook paper, scan/take a picture then submit it as an attachment where indicated. Provide as much work as possible to earn all points. If I can’t see your work clearly, I will not make assumptions. The test is worth 85points. The percent grade will be posted. First Order ODEs and IVPs Theorem: M(x,y) dx + N(x,y) dy = 0 is an exact equation if dM/dy = dN/dx. Exact Equations: dy/dx + P(x)y = Q(x) 10 pts each Classify each equation below as linear or exact. Then solve each problem. (ex + y) dx + (x – 2 – yey ) dy = 0 (y + x2 sin x) dx – x dy = 0 x dy/dx = 2y + x3 cos x, y(π2)=0 II. Mixing Application dx/dt = input rate(concentration in) – output rate (concentration out) concentration out = x / (volume + (in rate – out rate)t ) 20 pts The Love Dough Company makes delicious Valentine cookies and cookie dough ice cream to sell during the month of February. The thing that makes their cookie dough stand out from the rest is their secret ingredient! Initially there is 100 lbs of cookie dough mixture with no secret ingredient in a tank. The company adds 1 kg/lb of the secret ingredient in the mixture at a constant rate of 5 lbs\min. The mixture inside the tank is kept well stirred and it flows out of the tank at a rate of 4 lbs\min. This is set aside to go into the cookie dough ice cream! How much of the secret ingredient is in the tank at time t? How much special ingredient is in the take after 60 minutes? III. Homogeneous ODEs Solve the following. a. 7pts y ” +2y’ + 5y = 0 b. 8pts y ‘” + y ’’ – y ‘ – y = 0 IV. Nonhomogeneous ODEs For undetermined coefficients: yp = generic expression of f(t) 20pts Consider the differential equation given by ay ” + by’ + cy = f(t). Solve the initial value problem using undetermined coefficients: y ” – 6y’ + 9y = 9×2 – 3x +14 y(0)=0 y’(0)=0