Final Review Set (Summer 2017)- Math 246 Show all your work for each problem 1) [6] (a) Use Euler's Method to approximate the solution to x '(t) = 1+ t sin(tx), x(0) = 0 at t=1 using eight steps. (b) Do the same with the improved Euler method and compare results. 1 2) [6] Solve the initial value problem tan(y) 2 + x sec 2 (y) + y'(x) = 0; y(0) = 1 . y 3) [6] Find the general solution for the Bernoulli differential equation: y'(x) + 2y = x y2 4) [6] Solve the initial value problem y''(x) + 6y'(x) + 5y(x) = 0, y(0) = 1, y'(0) = 5) [6] Use a power series expansion about x=0 to find a general solution for: (2x 3)y'' xy'+y = 0 (Determine at least the first four nonzero terms.) 1 1 . 3 6) [8] A damped vibrating spring under an external driving force can be modeled by the equation: my''+by'+ky = g(t) where m>0 is the mass of the spring, b is the damping constant, k>0 the spring constant and g(t) the driving force. If y(t) is the displacement from equilibrium at time t, determine the form (general solution) of the equation of motion if g(t) = sin( t) (assume b 2 < 4mk ). What is the behavior of the solution as t ? 7) [7] Solve the initial value problem y''+ 2y'+ y = 0, y(0) = 1, y'(0) = 2 . 8) [6] Solve the problem y''' 4y'' y'+ 4y = 0 using matrix methods, that is: Convert the equation to a system of first order linear equations---that is, a matrix equation of the form: where, Ax =0 x1 x = x2 x3 2 9) [6] A 12 hour water clock (clepsydra) is designed to the dimesions shown in the figure, where the shape of the surface is obtained by revolving the curve = around the y-axis. What should this curve be and what should the radius of the (circular) bottom hole be so that the water rate falls at a constant rate of 4 inches per hour? [Hint: The rate of change of the vertical axis y is governed by Torricelli's Law, which leads to the equation: ! = 2 = where A(y) is the cross-sectional area at height y, and a is the (constant) area of the drain hole at the bottom, and g is the gravatational acceleration (constant). Note that these latter two constants can be combined into the constant k.] 10) [6] Solve the initial value problem: (!) 5 !! + 100 ! 500 = 0; 0 = 0, ! 0 = 10, !! 0 = 250 given that ! = !! is one particualr solution of the equation. 11) [6] Find the general solution in powers of x of the differential equation given below. State the resursion relation and radius of convergence. ! + 2 !! + 4 ! + 2 = 0 12) [6] Plot a phase plane diagram for the system: = (2 ) = ( 5) 3 mgf' + by + k1: = slit) lEriven tha'tgllt) = Sllkt] We first find the homogeneous solution of the above equation. Thisis given oym'lfl 'l' W + kill 2 [l Corresponding characteristic equation is mg? + b' + l = l] The discriminant isx = b2 4m s: l] [as given] Henceaa = 1:2 41m = o12,o: e 111 D= The roots of the auxilliar'j-r equation is therefore 2m And so the homogeneous solution is of the form The particular solution must be of the form '9' = A SiEAt] + B (Edit) Hence, y' = AA coleAt) 13 5111011) and y\" = )12Asi11[)1t) 1231:0501) Thus we have m (AAsinmt) A23 eggplant: [111112051211] AB sin[}.t]+] l:[Asi11[At]+ 5120515111) = 511111;) ' This gives a linear equation in A and B which always has a solution So the general solution is of the form _Lt 03?: _Lt . 03 . y = 18 in cos + 28 in 5111 + Asmt] + BoosUut] where 5115211415 are constants. It can be seen from the equation that as t _} DO: Elli] _*" ASilkt) 'l' BCD'Slht) as the terms _i at __b 1 of; e Emlcors () and e 3ml5111() 2m 2m tendtanskoo Thus, as i } DIG, the behaviour of the system approaches a sinusoidal curve. mgf' + by + k1: = slit) lEriven tha'tgllt) = Sllkt] We first find the homogeneous solution of the above equation. Thisis given oym'lfl 'l' W + kill 2 [l Corresponding characteristic equation is mg? + b' + l = l] The discriminant isx = b2 4m s: l] [as given] Henceaa = 1:2 41m = o12,o: e 111 D= The roots of the auxilliar'j-r equation is therefore 2m And so the homogeneous solution is of the form The particular solution must be of the form '9' = A SiEAt] + B (Edit) Hence, y' = AA coleAt) 13 5111011) and y\" = )12Asi11[)1t) 1231:0501) Thus we have m (AAsinmt) A23 eggplant: [111112051211] AB sin[}.t]+] l:[Asi11[At]+ 5120515111) = 511111;) ' This gives a linear equation in A and B which always has a solution So the general solution is of the form _Lt 03?: _Lt . 03 . y = 18 in cos + 28 in 5111 + Asmt] + BoosUut] where 5115211415 are constants. It can be seen from the equation that as t _} DO: Elli] _*" ASilkt) 'l' BCD'Slht) as the terms _i at __b 1 of; e Emlcors () and e 3ml5111() 2m 2m tendtanskoo Thus, as i } DIG, the behaviour of the system approaches a sinusoidal curve