Final answer:
1. To find the general solution of the first differential equation, assume a solution of the form y = t^r and solve for the roots. The general solution is y = c1t^(-1) + c2t^(-1/4). 2. To find the general solution of the second differential equation, find the complementary solution and a particular solution. The general solution is the sum of these two solutions. 3. To solve the system of ordinary differential equations, differentiate and rearrange the equations, substitute in the known values, and integrate to find the solutions. 4. To compute the Laplace transform of a function, use the integral formula and solve for F(s) in terms of s. The C-transform exists for s > 7. 5. To compute the inverse Laplace transform of a function and solve an initial value problem, use the method of integrating factors and solve for y(t).
Step-by-step explanation:
1. To find the general solution of the differential equation: 4t'y' + 8ty + y = 0, we can assume the solution takes the form y = t^r. Plugging this into the equation gives a quadratic equation for r. Solving for r gives the two roots -1 and -1/4, so the general solution is y = c1t^(-1) + c2t^(-1/4).
2. To find the general solution of the differential equation: y'' + 4y = 8cos(2t), we can use the fact that the complementary solution is given by y=c1cos(2t)+c2sin(2t). We then find a particular solution by assuming a solution of the form y_p = a*cos(2t)+b*sin(2t), substitute it into the equation, and solve for the coefficients a and b. The general solution is then the sum of the complementary solution and the particular solution.
3. To find the solution to the system of ordinary differential equations: Tx' = 4x+y+et and y = -2c+y, we can start by differentiating both equations. This gives T x'' = 4x'+y'+e and y' = -2c'+y'. Rearranging the equations and substituting in the known values, we can solve for x'' and y''. Then we integrate the equations twice to find x and y, and substitute in the known values.
4. To compute the Laplace transform F(s) = {f(t)}(s) of the function f(t) = e^7sin't, we use the formula for the Laplace transform of a function, which is defined as the integral from 0 to infinity of e^(-st)f(t) dt. In this case, we substitute in the function f(t) and solve the integral to find F(s) in terms of s. The C-transform exists for s > 7.
5. (a) To compute the inverse Laplace transform --'{F(s)}(t) of the function F(s) such that: s? F(a) – 4F (s) = v, we start by rearranging the equation to solve for F(s). Then we compute the inverse Laplace transform of F(s) to find the corresponding function f(t). (b) To find the solution to the initial value problem y' - 4y = 5sin(t), y(0) = 0, 7(0) = 0, we can use the method of integrating factors. We write the equation in the form y' + p(t)y = q(t), where p(t) = -4 and q(t) = 5sin(t). Then we find the integrating factor, multiply both sides of the equation by it, and integrate both sides to solve for y(t).