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Maths › Differential Equations ›

Differential Equations Worked Examples 3

Worked examples which include trigonometrical funvtionson the right hand side

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Overview
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Overview

In the following worked examples $\displaystyle A\;cos\,nx\;+\;B\;sin\,nx$ is usually re-written as $\displaystyle C\;sin\,(nx\;+\;\alpha )$ . For those unused to this type of trigonometrical manipulation, the following notes should help.

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Example 1:

$\ddot{x}\;+\;3\dot{x}\;+\;2x\;=\;4$

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$\therefore\;\;\;\;\;x\;=\;A\,e^{-2x}\;+\;B\,e^{-x}\;+\;\frac{4}{2}$

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Example 2:

$\mathbf{\ddot{x}\;+\;4\dot{x}\;+\;3x\;=\;15\;sin\,3t}$

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To find the Particular Integral:-

$Let\;\;\;\;\;x\;=\;A\;cos\,3t\;+\;B\;sin\,3t$

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$\therefore\;\;\;\;\;\dot{x}\;=\;-\,3A\;sin\,3t\;+\;3B\;cos\,3t$

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$\therefore\;\;\;\;\;\ddot{x}\;=\;-\,9A\;cos\,3t\;-\;9B\;sin\,3t$

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Substituting in the original equation.

$-\,9A\;cos\,3t\;-\;9B\;sin\,3t\;-12\,A\;sin\,3t\;+\;12\,B\;cos\,3t\;+\;3\,A\;cos\,3t\;+\;3\,B\;sin\,3t\;=\;5\;sin\,3t$

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Equating the coefficients of sin 3t and cos 3t

$(12B\;+\;3A\;-\;9A)\;cos\,3t\;=\;0$

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$(3B\;-\;9B\;-\;12A)\;sin\,3t\;=\;5\;sin\,3t$

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$\therefore\;\;\;\;\;12\;B\;=\;6\;A\;\;\;\;or\;\;\;\;A\;=\;2B$

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$and\;\;\;\;\;-\;6B\;-\;12A\;=\;5$

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$Thus\;\;\;\;\;\;x\;=\;-\;\frac{\sqrt{5}}{6}\;sin\,(3t\;+\;\alpha )$

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To find the Complementary Function using the D factor:-

$\ddot{x}\;+\;4\dot{x}\;+\;3x\;=\;0$

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$\therefore\;\;\;\;D^2\;+4D\;+\;3\;=\;0$

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$Thus\;\;\;\;D\;=\;-3\;\;\;\;or\;\;\;\;-1$

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$A\;e^{-t}\;+\;B\,e^{-3t}$

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And the General Solution is:-

$\mathbf{x\;=\;-\;\frac{\sqrt{5}}{6}\;sin\,(3t\;+\;\alpha )\;+\;A\;e^{-t}\;+\;B\,e^{-3t}\;\;\;\;where\;\;tan\,\alpha \;=\;2}$

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Example 3:

Find the General Solution to the following equation.

$\mathbf{\ddot{x}\;+\;9x\;=\;15\;cos\,4t}$

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And the General Solution is :-

$\mathbf{x\;=\;-\,\frac{\sqrt{5}}{6}\;cos\,4t\;+\;A\,cos\,3t\;+\;B\,sin\,3t}$

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Example 4:

$\mathbf{\ddot{y}\;+\;4\,\dot{y}\;+\;40y\;=\;500\;cos\,5t}$

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Thus the General Solution is :-

$\mathbf{y\;=\;20\,cos\,(5t\;-\;\alpha )\;+\;e^{-2t}\;(A\,cos\,6t\;+\;B\,sin\,6t)\;\;\;\;\;where\;tan\,\alpha \;=\;\frac{4}{3}}$

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Example 5:

Find the General Solution to :-

$\mathbf{\frac{d^2y}{dx^2}\;+\;6\,\frac{dy}{dx}\;+\;13y\;=\;10\,cos\;x}$

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Thus the General Solution is:-

$\mathbf{y\;=\;\frac{\sqrt{5}}{3}cos\,(x\;-\;\alpha )\;+\;e^{-3x}\,(A\,cos\,2x\;+\;B\,sin\,2x)\;\;\;\;where\;\;tan\,\alpha \;=\;\frac{1}{2}}$

(64)

Example 6:

Show that the equation $\displaystyle \ddot{x}\;+\;n^2\,x\;=\;a\,cos\,pt$ has a solution $\displaystyle \dot{x}\;=\;A\,cos\,pt\;when\;p\;\neq n$ and find A.
The equation of motion of a body is $\displaystyle \ddot{x}\;+\;x\;=\;6\,cos\,2t.$ Find the General Solution for x in terms of t and show that if $\displaystyle x\;=\;\dot{x}\;=\;0\;at\;t\;=\;0\;\;then\;\;x\;=\;2\,(cos\,t\;-\;cos\,2t)$ .
What is the largest displacement of the body (a) in a positive direction and (b) in a negative direction.

Example 7:

A body which weighs 16 lbs. is moving in a straight line and it is acted upon by the following forces when it is x feet from a fixed point O in the line.

2.5x lb.wt.towards O
A resistance of $\displaystyle 2\,\dot{x}\,lb.wt.$
A force of cos t lb.wt. in the positive direction.

Show that:-

$\ddot{x}\;+\;4\,\dot{x}\;+\;5x\;=\;2\,cos\,t$

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Hence find:-

The Steady State.
The Transient.
The General Solution.

And if

$If\;\;\;x\;=\;1\;\;\;and\;\;\;\dot{x}\;=\;0\;\;\;at\;\;\;t\;=\;0$

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Show That:-

$x\;=\;\frac{1}{2\sqrt{2}}\;cos\,(t\;-\;\frac{1}{4}\pi )\;+\;\frac{1}{4}\;e^{-2t}\;(3\,cos\,t\;+\;5\;sin\,t)$

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Example 8:

Show that $\displaystyle x\;=\;A\;e^{\lambda \,t}$ is a solution of the equation $\displaystyle \ddot{x}\;+\;2k\dot{x}\;+\;n^2\,x\;=\;a\,e^{\lambda t}$ if $\displaystyle A\left(\lambda ^2\;+\;2k\lambda \;+\;n^2 \right)\;=\;a$ and hence find the Complete solution of the equation $\displaystyle \ddot{x}\;+\;6\dot{x}\;+\;6x\;=\;8\,e^{-4t}$

Using the method of the above example find the General Solution to the following Equations,