MXB106 Take Home Assignment 4 Question 1 Consider a spherical object whose steady state temperature distribution is described by the equation 12r ddr r2 d dru(r) = 0 on the domain 1 < r < 2. Solve for the temperature distribution with boundary conditions u(1) = 0.5 and u0(2) = 0.5 and plot the temperature distribution over the relevant domain. Question 2 Consider the following non-homogeneous ODE y00(t) − 8y0(t) + 16y(t) = kd(t) where k is a constant and d(t) is the Dirac delta function. Solve this ODE using Laplace transforms with the boundary conditions y(0) = a and y0(0) = b. Question 3 Consider the second order differential equation y00(x) + p(x)y0(x) + q(x)y(x) = 0 Given that y(x) = y1(x) is a known solution to this ODE, find a second solution, y2(x) in terms of y1(x), p(x), and q(x). Question 4 Solve y00(x) + 9y(x) = 0 where y(0) + y0(0) = 0 and y(p) − y0(p) = 10. Plot the solution on the domain 0 < x < p. 1Question 5 Solve the following differential equation y00(x) + y0(x) = 1 1 + ex via variation of parameters. Question 6 Given that y1(x) and y2(x) satisfy the differential equation f1(x)y00(x) + f2(x)y0(x) + f3(x)y(x) = 0 show that y(x) = c1y1(x) + c2y2(x) also satisfies this differential equation. Question 7 Find the values of a such that the system ~ x0(t) = a 3 − −9 3~ x(t) will have periodic solutions. Question 8 Determine the asymptotic stability of the solution ~ x = (0, 0)T for the system ~ x0(t) = − c4 3 1~ x(t) for c = 5, 7, and 10. Question 9 Find the Laplace transform of tn f 0(t) where n ≥ 1. 2Question 10 The fundamental set solutions of a third order ODE whose characteristic polynomial has one real root and two complex roots is y1(t) = el1t, y2(t) = eat cos(bt), and y3(t) = eat sin(bt) where l1,2 = a ± bi. Prove that y1(t), y2(t), and y3(t) indeed form the fundamental set of solutions, i.e. prove they are linearly independent. 3