Can you show me step by step how you got the answer to these questions. Thank yo

Question

Can you show me step by step how you got the answer to these questions.

Thank you

Solve the differential equation dy/dx = y + 3/x - 1 That satisfies the initial condition y(x = 0)=1. The displacement, x (cm), of a particle at time t is given by d^2 x/dt^2 + 2 dx/dt + 5x = 4e Find the general solution. Find the particular solution given that the particle is at rest at the origin at t=0. The current I in the electrical circuit shown in the figure opposite satisfies the following differential equation: Where L = 1 mH is the inductance, R=200 ohm and C is the capacitance. For what values of C will the current be highly damped, oscillatory behaviour? Give the general solution of above equation (1) when: you may find it helpful to look through exit test question E5 with the answers (Module M6-3 in the PPLATO)

Explanation / Answer

1) this is variable seprable case

=> dy / (y+3) = dx/(x - 1) integrate both side

=> ln|(y+3|) = ln(|x-1|) + c

put x = 0 and y = 1

c = ln 3

=> (y+3)/(x-1) = 3 Answer

2) this is case of 2nd order non homogenious equation

So Characterstic eq wil be a^2 + 2a + 5 =0

=> roots = -2 + sqrt(-4*4)/2 = -1 + 2i and -1 - 2i

=> general solution = e^-t(c1*cos(2t) + c2*sin(2t)) where c1 and c2 are constants

particular solution

let yp = Ae^t

yp' = Ae^2

yp'' = Ae^t

=> (A + 2A + 5A)e^t = 4e^2

=> A = 0.5

=> Yp = 0.5e^t

So Final solution = e^-t(c1*cos(2t) + c2*sin(2t)) + 0.5*e^t Answer

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