Problem 1 (a) If z = exp(bt), where a and b are not functions of t,...
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
Convolution Integrals. For part A the solution I got was t*exp(z*t) and for part B the solution I got was (exp(z2*t) - exp(z1*t))/(z2-z1). I need help with the third part of the question calculating (f * f)(t) without computing any integrals. f(t -s)g(s)ds by hand for (a) and (b) below Calculate (a) f(t) g(t) = et where z is a constant e21t and g(t) e22t where z1 and z2 are constants (b) f(t) Use your results from parts (a) and...
The functions p(t) and g(t) are continuous for every t. It is stated that In(t+1) and sin(t) cannot both be solutions of the differential equation v +pg+qU 0. Which of the following imply this conclusion? A: The statement is incorrect. There exist a pair of everywhere continuous functions p(t) and g(t) that will make In(t+1) and sin(t) valid solutions B: The Wronskian of the two functions vanishes at but there are values oft where it does not vanish. C: If...
Previous Problem Problem List Next Problem (1 point) Note WeBWork will interpret acos(z) as cos (z), so in order to write a times cos(z) you need to type a cos(z) or put a space between them. The general solution of the homogeneous differential equation can be written as e-acos(x)+bsin(z) where a, b are arbitrary constants and 1r is a particular solution of the nonhomogeneous equation By superposition, the general solution of the equation y" + ly-2ez is บ-Uc + so...
Helpful Formulas: * Find the appropriate form of general solution for each of the following PDEs. (In cases, where 2. relevant find r) and w(r, 0) explicitly for the problem) (0,t)(7,t)0 Useful Formulas/Pacts for PDEs/Fourier Series L. n m) L, 2,2 cos ( z) dz = * Find the appropriate form of general solution for each of the following PDEs. (In cases, where 2. relevant find r) and w(r, 0) explicitly for the problem) (0,t)(7,t)0 Useful Formulas/Pacts for PDEs/Fourier Series...
use the hint please 2. Show that the Dirichlet problem for the disc t(z,y): +y S R2), where f(0) is the boundary function, has the solution 0o aj COS 1 sin j 3-1 where a, and b, are the Fourier coefficients of f. Show also that the Poisson integral formula for this more general disc setting is R22 (Hint: Do not solve this problem from first principles. Rather, do a change of variables to reduce this new problem to the...
Compute Laplace transforms of the following functions: (a) f1 = (1 + t) (b) f2 = eat sin(bt) 11, 0<t<1, (c) f3 = -1 1<t<2, | 2, t>2, Find the functions from their Laplace transforms: (a) Lyı] s(s + 1) (s +3) 2+s (b) L[42] = 52 + 2 s +5 (c) L[y3] = Solve the following initial value problems using the Laplace transform. Confirm each solution with a Matlab plot showing the function on the interval 0 <t<5. (a)...
57. Find the total derivative dz/dt, given (a) z = x^2− 8xy − y^3 , where x = 3t and y = 1 − t. (b) z = f(x, y, t), where x = a + bt, and y = c + k
please answer all the 4 parts of this question 2. Consider the circular helix r(t)- (a cos t, a sin t, bt) where a > 0,b > 0. Let P(0, a, T) be a point on the helix (a) Find the Frenet frame (T, N, B) at the point P (b) Find equations for the tangent and normal line at P (c) Find equations for the normal plane and the osculating plane at P (d) What is the curvature at...