Online exam on linear algebra and differential equation
Tutorial Sheet 6::: MTL 101::: Systems of ODEs and Laplace Transforms (1) Find the real general solution of the following systems. (a) y′1 = −8y1 − 2y2, y′2 = 2y1 − 4y2, (b) y′1 = −3y1 − y2 + 2y3, y′2 = −4y2 + 2y3, y′3 = y2 − 5y3, (c) y′1 = −y1 − 4y2 + 2y3, y′2 = 2y1 + 5y2 − y3, y′3 = 2y1 + 2y2 + 2y3. (2) Solve the following IVPs. (a) y′1 = 2y1 + 2y2, y ′ 2 = 5y1 − y2, y1(0) = 0, y2(0) = −7. (b) y′1 = −14y1 + 10y2, y′2 = −5y1 + y2, y1(0) = −1, y2(0) = 1. (3) Solve the following system of equations. (a) y′1 = y2 + e 3t, y′2 = y1 − 3e3t (b) y′1 = 3y1 + y2 − 3 sin 3t, y′2 = 7y1 − 3y2 + 9 cos 3t− 16 sin 3t (c) y′1 = −2y1 + y2, y′2 = −y1 + et. (4) Solve the following IVP. (a) y′1 = y2 − 5 sin t, y′2 = −4y1 + 17 cos t, y1(0) = 5, y2(0) = 2. (b) y′1 = y1 + 4y2 − t2 + 6t, y′2 = y1 + y2 − t2 + t− 1, y1(0) = 2, y2(0) = −1. (c) y′1 = 5y1 + 4y2 − 5t2 + 6t+ 25, y′2 = y1 + 2y2 − t2 + 2t+ 4, y1(0) = 0, y2(0) = 0. (5) Find the Laplace transform of the following functions. cos2wt, et cosh 3t, sin 2t cos 2t, e−αt cos βt, sinh t cos t, 2e−t cos2 1 2 t. (6) Find the inverse Laplace transform of the following functions. 5s s2−25 , 1−7s (s−3)(s−1)(s+2) , 2s3 s4−1 , 2 s2+s+ 1 2 (7) Solve the following IVP using Laplace transform. (a) y′′ − y′ − 2y = 0, y(0) = 8, y′(0) = 7. (b) y′′ + 2y′ − 3y = 6e−2t, y(0) = 2, y′(0) = −14. (8) Find the Laplace transform of the following functions (where u is the unit step function): tu(t− 1), e−2tu(t− 3), 4u(t− π) cos t. (9) Find the inverse Laplace transform of the following functions: e−3s/s3, 3(1− e−πs)/(s2 + 9), se−2s/(s2 + π2) (10) Solve the following IVP. (a) y′′ + 6y′ + 8y = e−3t − e−5t, y(0) = 0, y′(0) = 0. (b) y′′ + 3y′ + 2y = 4t if 0 < t="">< 1="" and="" 8="" if="" t=""> 1; y(0) = 0, y′(0) = 0. (c) y′′ + 4y′ + 5y = δ(t− 1), (δ is the Dirac’s Delta) y(0) = 0, y′(0) = 3. (d) y′′ + 5y′ + 6y = u(t− 1) + δ(t− 2) (where u, δ are the step function and the dirac’s Delta function), y(0) = 0 and y′(0) = 1. (11) Find the Laplace transform (by differentiation) of the following functions: t2 cosh πt, te−t sin t, t2 coswt (12) Find inverse Laplace transform of the following functions by differentiation or integration: 1 (s−3)3 , 2s+6 (s2+6s+10)2 , ln( s+a s+b ), cot−1 s π . (13) Compute convolution of the following: 1 ∗ sinwt, et ∗ e−1, coswt ∗ sinwt, u(t− 1) ∗ t2, u(t− 3) ∗ e2t. (14) Use convolution theorem to compute the inverse transform: 6 s(s+ 3) , s2 (s2 + w2)2 , e−as s(s+ s− 2) , w s2(s2 + w2) , 1 (s+ 3)(s− 2) (15) Solve IVP by using convolution. (a) y′′ + y = 3 cos 2t; y(0) = 0, y′(0) = 0. (b) y′′ + 2y′ + 2y = 5u(t− 2π) sin t; y(0) = 1, y′(0) = 0. (c) y′′ + y = r(t), r(t) = 4t if 1 < t="">< 2="" and="" 0="" otherwise;="" y(0)="0," ,="" y′(0)="0." (d)="" y′′="" +="" 3y′="" +="" 2y="r(t)," r(t)="4t" if="" 0="">< t="">< 1="" and="" 8="" if="" t=""> 1; y(0) = 0, y′(0) = 0. (16) Solve the integral equations using Laplace transform. y(t) = 1 + ∫ t 0 y(r) dr, y = 2t− 4 ∫ t 0 y(r)(t− r) dr, y(t) = 1− sinh t+ ∫ t 0 (1 + r)y(t− r) dr. 2 (17) Use partial fraction method to find the Laplace transform of the following: 6 (s+ 2)(s− 4) , s2 + 9s− 9 s3 − 9s , s3 + 6s2 + 14s (s+ 2)4 . (18) Derive the following formulae. (a) L−1{ 1 s4+4a4 } = 1 4a3 (cosh at sin at− sinh at cos at), (b) L−1{ s s4+4a4 } = 1 2a2 sinh at sin at, (19) Solve the following IVPs (using Laplace transform). (a) y′1 = −y1 + y2, y′2 = −y1 − y2, y1(0) = 1, y2(0) = 0, (b) y′′1 + y2 = −5 cos 2t, y′′2 + y1 = 5 cos 2t, y1(0) = 1, y′(0) = 1, y2(0) = −1, y′2(0) = 1. (c) y′1 = 2y1 + 4y2 + 64tu(t− 1), y′2 = y1 + 2y2, y1(0) = −4, y2(0) = −4. :: END :: mal101- notes copy.pdf 28 Tutorial Sheet 4::: MAL 101::: Differential Equations (1) Verify that f is a particular solution of the given (IVPs) Initial Value Problems: (a) dy dx = y − x, y(0) = 3, f(x) = 2ex + x+ 1. (b) dy dx = y tan x, y(0) = 1 2 π, f(x) = π 2 sec x. (2) Solve the following separable equations: (a) (x− 4)y4 dx− x3(y2 − 3) dy = 0; (b) x sin y dx+ (x2 + 1) cos y dy = 0. (c) y′ = x 2+y2 xy (Substitute y x = u); (d) y′ + cosec y = 0. (3) Solve the following equation by reducing it to a separable equation: (x2 − 3y2) dx+ 2xy dy = 0. (4) Determine whether or not each of the given equations is exact; solve those that are exact. (a) (6xy + 2y2 − 5) dx+ (3x2 + 4xy − 6) dy = 0; (b) (y2 + 1) cosx dx+ 2y sin x dy = 0; (c) (2xy + 1) dx+ (x2 + 4y) dy = 0; (d) (3x2y + 2) dx− (x3 + y) dy = 0. (e) −2xy sin(x2)dx+ cos(x2)dy = 0. (f) (e(x+y) − y)dx+ (xe(x+y) + 1)dy = 0 (5) Solve the IVPs. (a) (2xy − 3) dx+ (x2 + 4y) dy = 0, y(1) = 2; (b) (3x2y2 − y3 + 2x) dx+ (2x3y − 3xy2 + 1) dy = 0, y(−2) = 1. (c) y dy dx + 4x = 0, y(0) = 2. (d) dr dθ = b ( dr dθ cos θ + r sin θ ) , r(π/2) = π. (6) Falling Body: Consider a stone falling freely through the air. Assuming that the air resistance is negligible and the acceleration due to gravity g = 9.8 m/s2, construct the resulting ODE and find its solution, if the initial position is h0 and the initial velocity is v0. (7) Subsonic Flight: The efficiency of engines of subsonic airplanes depends on the air pressure and (usually) is maximum at 36000 ft. The rate of change of air pressure y′(x) is proportional to air pressure y(x) at height x. If y0 is the pressure at sea level and the pressure decreases to half at 18000 ft, then find the air pressure at 36000 ft. (8) Dryer: In a laundry dryer loss of moisture is directly proportional to moisture content of the laundry. If wet laundry loses one fourth of its moisture in the first 10 minutes, when will the laundry be 95% dry. (9) Find all the curves in the xy-plane whose tangents pass through the point (a, b). (10) Under what conditions on constants A,B,C, and D, is (Ax + By)dx + (Cx + Dy)dy = 0 is exact. Solve the equation when it is exact. (11) Determine the constant A such that the equation is exact, and solve the resulting exact equa- tion: (a) (x2 + 3xy)dx+ (Ax2 + 4y)dy = 0; (b) ( 1 x2 + 1 y2 )dx+ (Ax+1 y3 )dy = 0. (12) Determine the most general function N(x, y) such that the equation is exact: (a) (x2 + xy2) dx+N(x, y) dy = 0; (b) (x−2y−2 + xy−3) dx+N(x, y) dy = 0. (13) Consider the differential equation (y2 + 2xy) dx− x2 dy = 0. (a) Observe that this equation is not exact. Multiply the given equation through by yn, where n is an integer, and then determine n so that yn is an integrating factor of the given equation. Solve the resulting exact equation. (b) Show that y = 0 is a solution of the original nonexact equation but is not a solution of the essentially equivalent exact equation found in (a). 29 (14) Consider a differential equation of the form [y + xf(x2 + y2)] dx+ [yf(x2 + y2)− x] dy = 0. (a) Show that an equation of this form is not exact. (b) Show that 1/(x2 + y2) is an integrating factor of an equation of this form. (15) Use the result of the above exercise to solve the equation [y + x(x2 + y2)2] dx+ [y(x2 + y2)2 − x] dy = 0. (16) Find all solutions of the following equations: (a) y′ − 2y = 1; (b) y′ + y = ex; (c) y′ − 2y = x2 + x; (d) 3y′ + y = 2e−x; (e) y′ + 3y = ex. (17) Consider the equation y′ + (cosx)y = e− sinx. (a) Find the solution φ which satisfies φ(π) = π. (b) Show that any solution φ has the property that φ(πk)−φ(0) = πk, where k is any integer. (18) Solve the Bernoulli’s equations: (a) y′ − 2xy = xy2; (b) y′ + y = xy3. (19) Solve the following nonlinear ODEs. (a) y′ sin 2y + x cos 2y = 2x (b) 2yy′ + y2 sin x = sin x y(0) = √ 5 (20) Solve the following IVP: (x− 1)y′ = 2y, y(1) = 1 Explain the results in view of the theory of existence and uniqueness of IVPs. (21) Find all the initial conditions, such that corresponding IVP, with ODE, (x2 − 4x)y′ = (2x− 4)y has no solution, unique solution and more than one solutions. (22) Show that the Lipschitz condition is satisfied by the function | sin y|+ x at every point on the xy-plane though its partial derivative with respect to y does not exist on the line y = 0. (23) Apply Picard’s iteration method to the following problems. Do three steps of the iteration. (a) y′ = y, y(0) = 1; (b) y′ = x+ y, y(0) = −1. (24) Consider the following IVP: ydy =