Differential Equations 6 etch the graph of solutions with y

Differential Equations

6. etch the graph of solutions with y (0) and y (0) spring mass systems. (a) m 1, b 3 and k 1 1, 2 and k 1 (b) m (c) m 1, b 1 and k -1 to the following

Solution

General Equation of Spring Mass System is given by

my\'\' + by\' + ky = 0

(a) Plugging the given values

=> y\'\' + 3y\' + y = 0

Let y = e^rtbe the solution of the equation , therefore the characteristic equation becomes

=> r² + 3r + 1 = 0

=> r₁ = (-3 + 5)/2 and r₂ = (-3 + 5)/2

=> y(t) = c₁e^{(-3 + 5)t/2} + c₂e^{(-3 - 5)t/2}

y(0) = c₁ + c₂ = 1 and y\'(0) = (-3 + 5)c₁/2 + (-3 - 5)c₂/2 = -1 => c₁ = (5 + 5)/10 and c₂ = (5 - 5)/10

=> y(t) = ( (5 + 5)/10 )e^{(-3 + 5)t/2} + ( (5 - 5)/10 )e^{(-3 - 5)t/2}

(b) Plugging the given values

=> y\'\' + 2y\' + y = 0

Let y = e^rtbe the solution of the equation , therefore the characteristic equation becomes

=> r² + 2r + 1 = 0

=> r₁ = -1 and r₂ = -1

=> y(t) = c₁e^-t + c₂te^-t

y(0) = c₁ = 1 and y\'(0) = -c₁ + c₂ = -1 => c₁ = 1 and c₂ = 0

=> y(t) = e^-t

(c) Plugging the given values

=> y\'\' + y\' + y = 0

Let y = e^rtbe the solution of the equation , therefore the characteristic equation becomes

=> r² + r + 1 = 0

=> r₁ = (-1 + 3i)/2 and r₂ = (-1 - 3i)/2

=> y(t) = c₁e^{(-1 + 3i)t/2} + c₂e^{(-1 - 3i)t/2}

=> y(t) = c₁e^-t/2cos(3t/2) + c₂e^-t/2sin(3t/2)

y(0) = c₁ = 1 and y\'(0) = (-1/2)c₁ + (3/2)c₂ = -1 => c₁ = 1 and c₂ = -1/3

=> y(t) = e^-t/2cos(3t/2) + (-1/3)e^-t/2sin(3t/2)