Solve the first order homogeneous differential equation -

Question

Q1 Solve the first order homogeneous differential equation dy/dx = (2x - y - 4)/(x + 2y -5)

Q2 Let f(x, y) = e^5x siny, find the Taylor series expansion of the function f(x +1/4, y + Π/3) in the form = e^5x(A sin y+ B cos y)i.e. find A, B

Q3 If y = e^6x sinx using Leibnitz's theorem on repeated differentiation, or otherwise show that

d⁴y/dx^{4 =} e^6x{1081 sinx + 840cosx}

Q4 The spring-mass-damped system shown in Fig 1, has data m = 0 kg, k =1800 Nm^-1 c₁= 650 Ns m^-1  c₂ = 700 Ns m^-1 w =10 , p =14, and initial conditions x(0). x^.(0), = 0.15ms^-1

Using the relationship sin(wt) e^iwt -e^-iwt , and the differential operator, calculate the displacement from equilibrium position function x(t).