Answered: Differential equations Assuming zero…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Differential equations

Assuming zero initial conditions, use classical methods to find solutions for the following differential equations:

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(a)

$\frac{d x}{d t} + 5 x = 2 \cdot \cos (3 t) Substitute \frac{d x}{d t} with x' x' + 5 x = 2 \cos (3 t) The equation is in first order linear ODE form Find the integration factor : μ (t) Find the integrating factor μ (x), so that : μ (x) \cdot p (x) = μ' (x) μ (t) = e^{5 t} Put the equation in the form (μ (x) \cdot y)' = μ (x) \cdot q (x) : (e^{5 t} x)' = 2 \cos (3 t) e^{5 t} Solve (e^{5 t} x)' = 2 \cos (3 t) e^{5 t} If f^{'} (x) = g (x) then f (x) = \int g (x) dx e^{5 t} x = \int 2 \cos (3 t) e^{5 t} dt \int 2 \cos (3 t) e^{5 t} dt = 2 (\frac{3 e^{5 t} \sin (3 t)}{34} + \frac{5 e^{5 t} \cos (3 t)}{34}) + c_{1} e^{5 t} x = 2 (\frac{3 e^{5 t} \sin (3 t)}{34} + \frac{5 e^{5 t} \cos (3 t)}{34}) + c_{1} Isolate x : x = \frac{3 \sin (3 t) + 5 \cos (3 t)}{17} + \frac{c_{1}}{e^{5 t}} x = \frac{3 \sin (3 t) + 5 \cos (3 t)}{17} + \frac{c_{1}}{e^{5 t}}$