1 Find the complete solution for vt given dvdt 4vt32 with y

1 Find the complete solution for v(t) given dv/dt + 4v(t)-32 with y(o)-18 2. Find the complete solution for y(t) given y\" + 4y\' + 3y = 12 with the initial conditions y(0)-6 and y(0),-8

Solution

Solution:-

1.

dv/dt + 4 v = 32

dv/dt = 32 – 4 v

dv = 32 dt - 4 v dt

apply integration both side

dv = 32 dt - 4vdt

  v(t) = 32 t – 4 v(t) t + c

given v(0) = 18

then c =18

complete solution

v(t) = (32 t + 18)/(1 +4 t)   answer

Question (2).

Solution:-

y’’ + 4 y’ + 3 y = 12

   d²y/dx² + 4 dy/dt + y = 12

dy/dt + 4 y + yt – 12 t + c₁ =0                           eq. (1)

Given y’(0)= 8 , y(0) = 6

8 + 4 x 6 +0 – 0 + c₁ = 0

c₁ = 32

Again integrate equation (1)

y+ 4yt + yt²/2 – 12 t²/2 +c₁t + c₂ = 0

6 + 4 y x0 +0+ 0 + 0 +c₂ = 0

c₂ = 6

Complete solution

y(t) = ( 6 t² – 32 t – 6 )/ ( 1 + 4t + t²/2)