midterm - Department of Chemical Engineering Fall 2005...

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Unformatted text preview: Department of Chemical Engineering Fall 2005 University of California, Santa Barbara ChE 230A Mid-Term Exam Prepare solutions to the following problems. The exam is closed book with one page of your own handwritten notes. Show all of your work. Good Luck! (25 pts.) Problem 1. Using a power series, solve the differential equation: y’=2xy with the boundary condition y(0) = 10. Compare your result with the solution obtained by direct integration. (40 pts.) Problem 2. Solve the potential function u(xy) that satisfies the partial differential equation in the half—plane over the intervals indicated: With u(x,0) ={3: w’lxl’; and u(r) --) 0 as r —-) 00., where r =- (x2 +y2)”2- (35 lpts.) Problem 3. Consider the' mechanical system illustrated below and suppose that mass m " l, the spring 1 stiffness is k,‘ x = O'is the equilibrium position, and the driving force F (t) is that plotted below. The equation of motion is the second—order differential equation: 56 + la 2 F (t) (a) Express F (t) as a Fourier series. (b) Solve for the “response” x(t). F(t) Mechanical Oscillator Far) A ...
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