Math 293 Practice Prelim #3Spring 2000(The actual exam will not be as long as this one.)Integration formulas which may be useful:Zxsinax dx=-xcosaxa+sinaxa2Zxcosax dx=xsinaxa+cosaxa2Zx2sinax dx=-x2cosaxa+2xsinaxa2cosaxa3Zx2cosax dx=x2sinaxaxcosaxa2-2sinaxa31. Letf(x)=x2when-1≤x≤1, and letf(x) be periodic with period 2.(a) Graphfon the interval-3≤x≤3.(b) Isfeven, odd, or neither? Isfcontinuous?(c) For what values ofxdoes the Fourier series offactually converge tof(x)?(d) Compute the Fourier series off.2. Solve the following initial-boundary value problem for the heat equation. It is notnecessary to give any derivation or justiﬁcation, for this problem.ut=3uxx,u(0,t)=0(π,t(x,0) = 3 sinx+ sin 2x-14sin 5x.3. Consider the second order equationy00-2y0+λy= 0 with the boundary conditionsy(0)=0,y(π) = 0. Find all the eigenvaluesλfor which the boundary value problem hasnontrivial solutions, and ﬁnd the nontrivial solutions. For full credit you should considereach of the three casesλ>1,λ= 1, andλ<1.4. Consider the boundary value problem, which is
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This homework help was uploaded on 01/21/2008 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).