PG entrance exam CUET PG 2025 Mock Test Engineering Mathematics Differential Equations Higher Order Linear Differential Equations with Constant Coefficients
If the motion of the mass m is subjected to an additional force of resistance, proportional to the instantaneous velocity of the mass, say λ\(\frac{d x}{d t}\) produced by a damper the oscillations are said to be damped. The equation of motion of the mass m is given by
m\(\frac{d^{2}x}{dt^{2}}\) = mg − k(e + x) − λ\(\frac{dx}{dt}\) = −kx − λ\(\frac{dx}{dt}\) (Writing \(\frac{\lambda}{m}\) = 2p and \(\frac{k}{m}\) = w2).
The general solution when p > w is:
1
x = e−pt(C1 \(e^{\sqrt{p−wt}}\) + C2.\(e^{−\sqrt{p−wt}}\))
2
x = e−pt(C1 \(e^{\sqrt{p^{2}+w^{2}}t}\) + C2.\(e^{−\sqrt{p^{2}+w^{2}}t}\))
3
x = ept(C1 \(e^{\sqrt{p^{2}−w^{2}t}}\) + C2.\(e^{−\sqrt{p^{2}−w^{2}t}}\))
4
x = e−pt(C1 \(e^{\sqrt{p^{2}−w^{2}}t}\) + C2.\(e^{−\sqrt{p^{2}−w^{2}t}}\))