(a) (i) Use the substitution sin(u) = x + 3 to show that dx 12x + 7 -1 (x+3) V-2x² – 12x+7+c. sin - |-2x² (ii) In part (i), we could have used the substitution cos(u) = x+3 instead, but the form of the antiderivative would be different. Without doing any calculations, explain how this is accounted for in the final result. 1 (b) In a given situation, the rate of change of the variable x is proportional to V16 – t2" (i) Using k as the constant of proportionality, write down a differential equation that describes this situation. (ii) Find the general solution of the differential equation you wrote down in (b)(i). You will need to make use of a (simple!) trigonometric substitution while solving this equation.

(a) (i) Use the substitution sin(u) = x + 3 to show that dx 12x + 7 -1 (x+3) V-2x² – 12x+7+c. sin - |-2x² (ii) In part (i), we could have used the substitution cos(u) = x+3 instead, but the form of the antiderivative would be different. Without doing any calculations, explain how this is accounted for in the final result. 1 (b) In a given situation, the rate of change of the variable x is proportional to V16 – t2" (i) Using k as the constant of proportionality, write down a differential equation that describes this situation. (ii) Find the general solution of the differential equation you wrote down in (b)(i). You will need to make use of a (simple!) trigonometric substitution while solving this equation.

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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

5
sin(u) = x + 3 to show that
V2
(a) (i) Use the substitution
3
sin-1
2
++3)-
dx =
(x+3)
V-2x2 –
12x +7+ c.
12x + 7
-
(ii) In part (i), we could have used the substitution cos(u) = x+3 instead, but the form
of the antiderivative would be different. Without doing any calculations, explain how
this is accounted for in the final result.
1
(b) In a given situation, the rate of change of the variable x is proportional to
V16 – t²
(i) Using k as the constant of proportionality, write down a differential equation that
describes this situation.
(ii) Find the general solution of the differential equation you wrote down in (b)(i). You
will need to make use of a (simple!) trigonometric substitution while solving this
equation.

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