To determine The critical motion y ( t ) = ( c 1 + c 2 t ) e − α t that starts from y 0 with initial velocity v 0 . Explanation Result used: Suppose the auxiliary equation is in the form m y ″ + c y ′ + k y = 0 , then Case ( i ) : If the auxiliary equation has distinct real roots λ 1 , and λ 2 of y = e λ x , then c 2 > 4 m k , that is, overdamping. Case ( ii ) : If the auxiliary equation has real double root λ = a of y = e λ x , then c 2 = 4 m k , that is, critical damping. Case ( iii ) : If the auxiliary equation has complex conjugate root λ 1 = a + i ω , λ 2 = a − i ω of y = e λ x , then then c 2 < 4 m k , that is, underdamping”. Calculation: It is known that, the damped mass-spring system can be modeled as, m y ″ + c y ′ + k y = 0 (1) Compute the auxiliary equation of the above ODE as follows. Let y = e λ x . That implies, the first and second derivative of y is y ′ = λ e λ x and y ″ = λ 2 e λ x , respectively. Substitute y ″ = λ 2 e λ x and y = e λ x in (1) and simplify further. m λ 2 e λ x + c λ e λ x + k e λ x = 0 e λ x ( m λ 2 + c λ + k ) = 0 m λ 2 + c λ + k = 0 λ 2 + c m λ + k m = 0 That is, the characteristic equation is λ 2 + c m λ + k m = 0 . Compute the roots of the above characteristic equation as follows. Substitute a = 1 and b = c m and c = k m in λ = − b ± b 2 − 4 a c 2 a To determine To graph: The solution curves for y 0 = 1 and α = 1 , for the given conditions.

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ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2.4, Problem 13P

To determine

The critical motion y(t)=(c1+c2t)e−αt that starts from y0 with initial velocity v0.

To determine

To graph: The solution curves for y0=1 and α=1, for the given conditions.

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Chapter 2.4, Problem 12P

Chapter 2.4, Problem 14P

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