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# Ordinary Differential Equation MATH 335

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This 1 page Class Notes was uploaded by Dr. Cleo Wunsch on Thursday October 15, 2015. The Class Notes belongs to MATH 335 at New Mexico Institute of Mining and Technology taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/223630/math-335-new-mexico-institute-of-mining-and-technology in Mathematics (M) at New Mexico Institute of Mining and Technology.

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Date Created: 10/15/15

to LO g cm Review for Test 1 Math 335 Applied Analysis I Fall 2001 Professor Rakhim Aitbayev Solve the differential equations a yy y 1sin3z b datdt 32 21 C 1 y2da y1 23dy 0 Solve the initial value problems a 1 3y y cos 3 y0 1 b yy 4302 1 310 1 c 32 1y 8mg 23 M0 1 Consider the equation y 1arx2 y6 y at 7 0 Determine for which of the initial conditions a y 0 0 b y1 0 and c y1 7 the IVP is guaranteed to have a unique solution a The half life of a certain isotope is 0 25 10 1 If a sample of radium weighs 10 9 now determine its weight in 100 years b A tank originally contains 2 lb of salt in 100 gal of water Brine containing 01 lbgal of salt is pumped into the tank at the rate of 10 galmin while the well mixed solution is drained off at 5 galmin Find the number of pounds of salt in the tank at any time before it over ows A a w Pure water is being pumped at the constant rate of 2 galmin into a 100 gal tank lled with a solu tion of water containing 10 gal of chemical while maintaining a constant volume and homogenous mixture Find a the number of gallons of chem ical at any time b the time when the solution has been diluted to one half the original amount A a w The number Nt of people in a community who are exposed to a particular advertisement is gov erned by he logistic equation Initially N0 500 and it is observed that N1 1000 If it is predicted that the limiting number of people in the community who will the advertisement is 50 000 determine Nt at any time Show that the functions yl 34 and y2 at lf form a fundamental set of solutions of the differen tial equation a 242gquot 5aty y 0 a gt 0 b Show that the function y clat 1 Cgat lf is a general solution of the differential equation 232yquot 5aty y 0 a gt 0 c Show that the two parameter family of functions y cm 6232 is a general solution of the differen tial equation at2y 2aty39 2y 0 a gt 0 6 Use the method of undetermined coefficients to solve the initial value problems a y 3y 2y 0 y0 0 y 0 1 b y y 0 y0 1 y 0 0 c y y 2y 0 y0 0 y 0 1 d 2y 2y y 0 y0 1y 00 e y 8y 16y 0 y0 2 y 0 1 7 Without solving the problem determine an interval in which the solution of the initial value problem In ty y cot t 0 y2 3 is certain to exist 8 Determine the critical points of the differential equation y y24 yf and classify each one as asymptotically stable unstable or semistable Answers 1a y ln y 1 13cos31 C y 1 1b lnz 21 2t C a 0 1c 12a1 yf C y i1 2a 1113 1 sina1 3 2b x1y2 2a2 v5 2c ya14 54112quot 3 a no b yes c no a m 4b mt t2 4 802t 40 4c mt 106 0 2 t 501n2 34 7min 4d Nt 500001 995 703 5 Prove 1 yl and y2 are solutions of the DE 2 yl and y2 are linearly independent 6a yt 6 2 6b yy cost 60 W 13t 6 6d yt e 2 cost2 6quot sint2 6e yt 26 Qte 7 17r 8 y2 assymt stable y 0 semistable y 2 unstable g

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